diff --git a/Combinatorial_map/include/CGAL/Cell_attribute.h b/Combinatorial_map/include/CGAL/Cell_attribute.h
index 713a2775c44..2a810e67d07 100644
--- a/Combinatorial_map/include/CGAL/Cell_attribute.h
+++ b/Combinatorial_map/include/CGAL/Cell_attribute.h
@@ -110,10 +110,10 @@ namespace CGAL {
     }
 
     /// Get the dart associated with the cell.
-    Dart_handle dart() { return NULL; }
+    Dart_handle dart() { return Refs::null_handle; }
 
     /// Get the dart associated with the cell.
-    Dart_const_handle dart() const { return NULL; }
+    Dart_const_handle dart() const { return Refs::null_handle; }
 
     /// Set the dart associated with the cell.
     void set_dart(Dart_handle) {}
@@ -234,7 +234,7 @@ namespace CGAL {
     /// Test if the cell is valid.
     /// A cell is valid if its dart is not NULL.
     bool is_valid() const
-    { return mdart!=NULL; }
+    { return mdart!=Refs::null_handle; }
 
     bool operator==(const Cell_attribute_without_info&) const
     { return true; }
@@ -244,7 +244,7 @@ namespace CGAL {
 
     //  protected:
     /// Contructor without parameter.
-    Cell_attribute_without_info() : mdart(NULL),
+    Cell_attribute_without_info() : mdart(Refs::null_handle),
                                     mrefcounting(0)
     {}
 
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map.h b/Combinatorial_map/include/CGAL/Combinatorial_map.h
index d5c5832c56d..4b20274e0f1 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map.h
@@ -96,7 +96,9 @@ namespace CGAL {
 
     static const size_type NB_MARKS = Base::NB_MARKS;
     static const unsigned int dimension = Base::dimension;
-    static const typename Base::Null_handle_type null_handle;
+
+    typedef typename Base::Null_handle_type Null_handle_type;
+    static Null_handle_type null_handle;
 
     using Base::null_dart_handle;
     using Base::mdarts;
@@ -2100,7 +2102,7 @@ namespace CGAL {
      */
     void topo_unsew_1(Dart_handle adart)
     {
-      CGAL_assertion( adart!=NULL && !adart->template is_free<1>() );
+      CGAL_assertion( !is_free<1>(adart) );
 
       int m = get_new_mark();
       std::deque<Dart_handle> dartv;
@@ -2133,7 +2135,7 @@ namespace CGAL {
      */
     void topo_unsew_0(Dart_handle adart)
     {
-      CGAL_assertion( adart!=NULL && !adart->template is_free<0>() );
+      CGAL_assertion( !is_free<0>(adart) );
       topo_unsew_1( adart->template beta<0>() );
     }
 
@@ -2147,7 +2149,7 @@ namespace CGAL {
     template<unsigned int i>
     void topo_unsew_for_involution(Dart_handle adart)
     {
-      CGAL_assertion( adart!=NULL && !adart->template is_free<i>() );
+      CGAL_assertion( !is_free<i>(adart) );
       CGAL_assertion( 2<=i && i<=Self::dimension );
 
       for ( CGAL::CMap_dart_iterator_of_involution<Self,i> it(*this, adart);
@@ -3209,9 +3211,9 @@ namespace CGAL {
       Dart_basic_range(Self &amap) : mmap(amap)
       {}
       iterator begin() { return iterator(mmap); }
-      iterator end()   { return iterator(mmap,NULL); }
+      iterator end()   { return iterator(mmap,mmap.null_handle); }
       const_iterator begin() const { return const_iterator(mmap); }
-      const_iterator end() const   { return const_iterator(mmap,NULL); }
+      const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
       size_type size()
       { return mmap.number_of_darts(); }
       bool empty() const
@@ -3226,7 +3228,7 @@ namespace CGAL {
       Dart_basic_const_range(Self &amap) : mmap(amap)
       {}
       const_iterator begin() const { return const_iterator(mmap); }
-      const_iterator end() const   { return const_iterator(mmap,NULL); }
+      const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
       size_type size() const
       { return mmap.number_of_darts(); }
       bool empty() const
@@ -3281,9 +3283,9 @@ namespace CGAL {
       One_dart_per_cell_range(Self &amap) : mmap(amap), msize(0)
       {}
       iterator begin() { return iterator(mmap); }
-      iterator end()   { return iterator(mmap,NULL); }
+      iterator end()   { return iterator(mmap,mmap.null_handle); }
       const_iterator begin() const { return const_iterator(mmap); }
-      const_iterator end() const   { return const_iterator(mmap,NULL); }
+      const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
       size_type size()
       {
         if (msize==0)
@@ -3306,7 +3308,7 @@ namespace CGAL {
       One_dart_per_cell_const_range(const Self &amap) : mmap(amap), msize(0)
       {}
       const_iterator begin() const { return const_iterator(mmap); }
-      const_iterator end() const   { return const_iterator(mmap,NULL); }
+      const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
       size_type size()
       {
         if (msize==0)
@@ -3371,13 +3373,13 @@ namespace CGAL {
      * simultaneously through all the darts of the two maps and we have
      * each time of the iteration two "dual" darts.
      */
-    Dart_handle dual(Self& amap, Dart_handle adart=NULL)
+    Dart_handle dual(Self& amap, Dart_handle adart=null_handle)
     {
       CGAL_assertion( is_without_boundary(dimension) );
 
       CGAL::Unique_hash_map< Dart_handle, Dart_handle,
         typename Self::Hash_function > dual;
-      Dart_handle d, d2, res = NULL;
+      Dart_handle d, d2, res = amap.null_handle;
 
       // We clear amap. TODO return a new amap ?
       amap.clear();
@@ -3387,7 +3389,7 @@ namespace CGAL {
             it!=darts().end(); ++it)
       {
         dual[it] = amap.create_dart();
-        if ( it==adart && res==NULL ) res = dual[it];
+        if ( it==adart && res==amap.null_handle ) res = dual[it];
       }
 
       // Then we link the darts by using the dual formula :
@@ -3422,7 +3424,7 @@ namespace CGAL {
 
       //  CGAL_postcondition(amap2.is_valid());
 
-      if ( res==NULL ) res = amap.darts().begin();
+      if ( res==amap.null_handle ) res = amap.darts().begin();
       return res;
     }
 
@@ -3669,7 +3671,7 @@ namespace CGAL {
 
   template < unsigned int d_, class Refs, class Items_, class Alloc_,
              class Storage_ >
-  const typename Combinatorial_map_base<d_,Refs,Items_,Alloc_,Storage_>::
+  typename Combinatorial_map_base<d_,Refs,Items_,Alloc_,Storage_>::
      Base::Null_handle_type
      Combinatorial_map_base<d_,Refs,Items_,Alloc_,Storage_>::null_handle =
      Combinatorial_map_base<d_,Refs,Items_,Alloc_,Storage_>::Base::null_handle;
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_constructors.h b/Combinatorial_map/include/CGAL/Combinatorial_map_constructors.h
index db31b5075f5..e1b0bf54b87 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_constructors.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_constructors.h
@@ -78,7 +78,6 @@ bool is_face_combinatorial_polygon(const Map& amap,
                                    unsigned int alg)
 {
   CGAL_assertion(alg>0);
-  CGAL_assertion(adart!=NULL);
 
   unsigned int nb = 0;
   typename Map::Dart_const_handle cur = adart;
@@ -127,8 +126,6 @@ template < class Map >
 bool is_volume_combinatorial_tetrahedron(const Map& amap,
                                          typename Map::Dart_const_handle d1)
 {
-  CGAL_assertion(d1!=NULL);
-
   typename Map::Dart_const_handle d2 = amap.beta(d1, 2);
   typename Map::Dart_const_handle d3 = amap.beta(d2, 0, 2);
   typename Map::Dart_const_handle d4 = amap.beta(d2, 1, 2);
@@ -232,8 +229,6 @@ template < class Map >
 bool is_volume_combinatorial_hexahedron(const Map& amap,
                                         typename Map::Dart_const_handle d1)
 {
-  CGAL_assertion(d1!=NULL);
-
   typename Map::Dart_const_handle d2 = amap.beta(d1, 1, 1, 2);
   typename Map::Dart_const_handle d3 = amap.beta(d2, 1, 1, 2);
   typename Map::Dart_const_handle d4 = amap.beta(d3, 1, 1, 2);
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_insertions.h b/Combinatorial_map/include/CGAL/Combinatorial_map_insertions.h
index 2267f0265e5..360466c83c8 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_insertions.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_insertions.h
@@ -35,7 +35,7 @@ template<class CMap>
 typename CMap::Dart_handle
 insert_cell_0_in_cell_1( CMap& amap, typename CMap::Dart_handle adart,
                          typename CMap::template
-                         Attribute_handle<0>::type ah=NULL )
+                         Attribute_handle<0>::type ah=CMap::null_handle )
 {
   typename CMap::Dart_handle d1, d2;
   int mark=amap.get_new_mark();
@@ -112,14 +112,14 @@ template < class CMap >
 typename CMap::Dart_handle
 insert_cell_0_in_cell_2( CMap& amap, typename CMap::Dart_handle adart,
                          typename CMap::template
-                         Attribute_handle<0>::type ah=NULL )
+                         Attribute_handle<0>::type ah=CMap::null_handle )
 {
-  CGAL_assertion(adart != NULL && adart!=amap.null_dart_handle);
+  CGAL_assertion(adart!=amap.null_dart_handle);
 
-  typename CMap::Dart_handle first=adart, prev=NULL, cur=NULL;
-  typename CMap::Dart_handle next=NULL, n1=NULL, n2=NULL;
-
-  typename CMap::Dart_handle nn1 = NULL, nn2 = NULL;
+  typename CMap::Dart_handle first=adart, prev=amap.null_handle,
+    cur=amap.null_handle, next=amap.null_handle,
+    n1=amap.null_handle, n2=amap.null_handle,
+    nn1=amap.null_handle, nn2=amap.null_handle;
 
   // If the facet is open, we search the dart 0-free
   while ( !amap.template is_free<0>(first) &&
@@ -145,21 +145,21 @@ insert_cell_0_in_cell_2( CMap& amap, typename CMap::Dart_handle adart,
       n1=amap.create_dart();
       amap.link_beta_0(cur, n1);
     }
-    else n1 = NULL;
+    else n1 = amap.null_handle;
 
     if (!amap.template is_free<1>(cur))
     {
       n2 = amap.create_dart();
       amap.link_beta_1(cur, n2);
     }
-    else n2 = NULL;
+    else n2 = amap.null_handle;
 
-    if ( n1!=NULL )
+    if ( n1!=amap.null_handle )
     {
-      if ( n2!=NULL )
+      if ( n2!=amap.null_handle )
         amap.basic_link_beta_0(n1, n2);
 
-      if ( prev!=NULL )
+      if ( prev!=amap.null_handle )
         amap.template basic_link_beta_for_involution<2>(prev, n1);
 
       CGAL::internal::Set_i_attribute_of_dart_functor<CMap, 0>::
@@ -172,15 +172,15 @@ insert_cell_0_in_cell_2( CMap& amap, typename CMap::Dart_handle adart,
       {
         if ( !amap.is_marked(amap.beta(cur, dim), treated) )
         {
-          if (n1!=NULL)
+          if (n1!=amap.null_handle)
           {
             nn1=amap.create_dart();
             amap.link_beta_1(amap.beta(cur, dim), nn1);
             amap.basic_link_beta_for_involution(n1, nn1, dim);
           }
-          else nn1=NULL;
+          else nn1=amap.null_handle;
 
-          if (n2!=NULL)
+          if (n2!=amap.null_handle)
           {
             nn2=amap.create_dart();
             amap.link_beta_0(amap.beta(cur, dim), nn2);
@@ -188,12 +188,12 @@ insert_cell_0_in_cell_2( CMap& amap, typename CMap::Dart_handle adart,
             CGAL::internal::Set_i_attribute_of_dart_functor<CMap, 0>::
                 run(&amap, nn2, ah);
           }
-          else nn2=NULL;
+          else nn2=amap.null_handle;
 
-          if (nn1 != NULL && nn2 != NULL)
+          if (nn1 != amap.null_handle && nn2 != amap.null_handle)
             amap.basic_link_beta_1(nn1, nn2);
 
-          if (nn1 != NULL && prev != NULL)
+          if (nn1 != amap.null_handle && prev != amap.null_handle)
             amap.template basic_link_beta_for_involution<2>
               (nn1, amap.beta(prev, dim));
 
@@ -201,10 +201,10 @@ insert_cell_0_in_cell_2( CMap& amap, typename CMap::Dart_handle adart,
         }
         else
         {
-          if ( n1!=NULL )
+          if ( n1!=amap.null_handle )
             amap.basic_link_beta_for_involution(n1,
                                                 amap.beta(cur, dim, 1), dim);
-          if ( n2!=NULL )
+          if ( n2!=amap.null_handle )
             amap.basic_link_beta_for_involution(n2,
                                                 amap.beta(cur, dim, 0), dim);
         }
@@ -216,7 +216,7 @@ insert_cell_0_in_cell_2( CMap& amap, typename CMap::Dart_handle adart,
   }
   while(cur!=first && cur!=amap.null_dart_handle);
 
-  if (n2 != NULL)
+  if (n2 != amap.null_handle)
   {
     amap.template basic_link_beta_for_involution<2>
         (amap.template beta<0>(first), n2);
@@ -265,7 +265,7 @@ typename CMap::Dart_handle
 insert_dangling_cell_1_in_cell_2( CMap& amap,
                                   typename CMap::Dart_handle adart1,
                                   typename CMap::template
-                                  Attribute_handle<0>::type ah=NULL )
+                                  Attribute_handle<0>::type ah=CMap::null_handle )
 {
   int mark1 = amap.get_new_mark();
   std::deque<typename CMap::Dart_handle> to_unmark;
@@ -278,8 +278,8 @@ insert_dangling_cell_1_in_cell_2( CMap& amap,
     }
   }
 
-  typename CMap::Dart_handle d1 = NULL;
-  typename CMap::Dart_handle d2 = NULL;
+  typename CMap::Dart_handle d1 = amap.null_handle;
+  typename CMap::Dart_handle d2 = amap.null_handle;
   unsigned int s1 = 0;
 
   int treated=amap.get_new_mark();
@@ -386,7 +386,7 @@ insert_cell_1_in_cell_2(CMap& amap,
                         typename CMap::Dart_handle adart1,
                         typename CMap::Dart_handle adart2)
 {
-  if ( adart2==NULL ) return insert_dangling_cell_1_in_cell_2(amap,adart1);
+  if ( adart2==amap.null_handle ) return insert_dangling_cell_1_in_cell_2(amap,adart1);
 
   CGAL_assertion(is_insertable_cell_1_in_cell_2<CMap>(amap, adart1, adart2));
 
@@ -407,8 +407,8 @@ insert_cell_1_in_cell_2(CMap& amap,
     }
   }
 
-  typename CMap::Dart_handle d1=NULL;
-  typename CMap::Dart_handle d2=NULL;
+  typename CMap::Dart_handle d1=amap.null_handle;
+  typename CMap::Dart_handle d2=amap.null_handle;
   unsigned int s1=0;
 
   int treated=amap.get_new_mark();
@@ -511,20 +511,20 @@ bool is_insertable_cell_2_in_cell_3(const CMap& amap,
 
   // The path must have at least one dart.
   if (afirst==alast) return false;
-  typename CMap::Dart_const_handle prec = NULL;
-  typename CMap::Dart_const_handle od = NULL;
+  typename CMap::Dart_const_handle prec = amap.null_handle;
+  typename CMap::Dart_const_handle od = amap.null_handle;
 
   for (InputIterator it(afirst); it!=alast; ++it)
   {
     // The path must contain only non empty darts.
-    if (*it == NULL || *it==amap.null_dart_handle) return false;
+    if (*it == amap.null_handle || *it==amap.null_dart_handle) return false;
 
     // Two consecutive darts of the path must belong to two edges
     // incident to the same vertex of the same volume.
-    if (prec != NULL)
+    if (prec != amap.null_handle)
     {
       od = amap.other_extremity(prec);
-      if ( od==NULL ) return false;
+      if ( od==amap.null_handle ) return false;
 
       // of and *it must belong to the same vertex of the same volume
       if ( !CGAL::belong_to_same_cell<CMap, 0, 2>(amap, od, *it) )
@@ -535,7 +535,7 @@ bool is_insertable_cell_2_in_cell_3(const CMap& amap,
 
   // The path must be closed.
   od = amap.other_extremity(prec);
-  if ( od==NULL ) return false;
+  if ( od==amap.null_handle ) return false;
 
   if (!CGAL::belong_to_same_cell<CMap, 0, 2>(amap, od, *afirst))
     return false;
@@ -555,7 +555,8 @@ insert_cell_2_in_cell_3(CMap& amap, InputIterator afirst, InputIterator alast)
 {
   CGAL_assertion(is_insertable_cell_2_in_cell_3(amap,afirst,alast));
 
-  typename CMap::Dart_handle prec = NULL, d = NULL, dd = NULL, first = NULL;
+  typename CMap::Dart_handle prec = amap.null_handle, d = amap.null_handle,
+    dd = amap.null_handle, first = amap.null_handle;
   bool withBeta3 = false;
 
   {
@@ -572,7 +573,7 @@ insert_cell_2_in_cell_3(CMap& amap, InputIterator afirst, InputIterator alast)
       if ( withBeta3 )
         dd = amap.create_dart();
 
-      if (prec != NULL)
+      if (prec != amap.null_handle)
       {
         amap.basic_link_beta_0(prec, d);
         if (withBeta3)
@@ -604,8 +605,8 @@ insert_cell_2_in_cell_3(CMap& amap, InputIterator afirst, InputIterator alast)
   {
     if ( !amap.is_free(first, dim) )
     {
-      typename CMap::Dart_handle first2 = NULL;
-      prec = NULL;
+      typename CMap::Dart_handle first2 = amap.null_handle;
+      prec = amap.null_handle;
       for ( CMap_dart_iterator_basic_of_orbit<CMap, 1> it(amap, first);
             it.cont(); ++it )
       {
@@ -618,7 +619,7 @@ insert_cell_2_in_cell_3(CMap& amap, InputIterator afirst, InputIterator alast)
             (amap.template beta<2,3>(it), dd, dim);
           amap.template basic_link_beta_for_involution<3>(d, dd);
         }
-        if ( prec!=NULL )
+        if ( prec!=amap.null_handle )
         {
           amap.link_beta_0(prec, d);
           if ( withBeta3 )
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_iterators_base.h b/Combinatorial_map/include/CGAL/Combinatorial_map_iterators_base.h
index 12e8fcd9aae..5f94af54381 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_iterators_base.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_iterators_base.h
@@ -206,7 +206,7 @@ namespace CGAL {
       mmark_number(amark),
       minitial_dart(adart)
     {
-      if ( minitial_dart!=NULL )
+      if ( minitial_dart!=amap.null_handle )
       {
         this->mmap->mark_null_dart(mmark_number);
         this->mmap->mark(minitial_dart, mmark_number);
@@ -320,7 +320,7 @@ namespace CGAL {
     CMap_extend_iterator(Map& amap, Dart_handle adart, int amark):
       Base(amap, adart, amark)
     {
-      if ( this->minitial_dart!=NULL &&
+      if ( this->minitial_dart!=amap.null_handle &&
            !this->mmap->is_free(this->minitial_dart, Bi) &&
            this->mmap->beta(this->minitial_dart, Bi)!=this->minitial_dart )
       {
@@ -492,9 +492,9 @@ namespace CGAL {
       mmap(amap), mdart(adart), msize(0)
     {}
     iterator begin()             { return iterator(mmap,mdart); }
-    iterator end()               { return iterator(mmap,NULL); }
+    iterator end()               { return iterator(mmap,mmap.null_handle); }
     const_iterator begin() const { return const_iterator(mmap,mdart); }
-    const_iterator end() const   { return const_iterator(mmap,NULL); }
+    const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
     typename Map_::size_type size()
     {
       if (msize==0)
@@ -520,9 +520,9 @@ namespace CGAL {
       mmap(amap), mdart(adart), msize(0), mmark(amark)
     {}
     iterator begin()             { return iterator(mmap,mdart,mmark); }
-    iterator end()               { return iterator(mmap,NULL,mmark); }
+    iterator end()               { return iterator(mmap,mmap.null_handle,mmark); }
     const_iterator begin() const { return const_iterator(mmap,mdart,mmark); }
-    const_iterator end() const   { return const_iterator(mmap,NULL,mmark); }
+    const_iterator end() const   { return const_iterator(mmap,mmap.null_handle,mmark); }
     typename Map_::size_type size()
     {
       if (msize==0)
@@ -549,7 +549,7 @@ namespace CGAL {
       mmap(amap), mdart(adart), msize(0)
     {}
     const_iterator begin() const { return const_iterator(mmap,mdart); }
-    const_iterator end() const   { return const_iterator(mmap,NULL); }
+    const_iterator end() const   { return const_iterator(mmap,mmap.null_handle); }
     typename Map_::size_type size()
     {
       if (msize==0)
@@ -574,7 +574,7 @@ namespace CGAL {
       mmap(amap), mdart(adart), msize(0), mmark(amark)
     {}
     const_iterator begin() const { return const_iterator(mmap,mdart,mmark); }
-    const_iterator end() const   { return const_iterator(mmap,NULL,mmark); }
+    const_iterator end() const   { return const_iterator(mmap,mmap.null_handle,mmark); }
     typename Map_::size_type size()
     {
       if (msize==0)
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h b/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
index 67a0096ba34..2c73b27e7af 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_operations.h
@@ -95,7 +95,7 @@ namespace CGAL
       size_t res = 0;
 
       typename CMap::Dart_handle d1, d2;
-      typename CMap::Dart_handle dg1=NULL, dg2=NULL;
+      typename CMap::Dart_handle dg1=amap.null_handle, dg2=amap.null_handle;
 
       int mark = amap.get_new_mark();
       int mark_modified_darts = amap.get_new_mark();
@@ -109,14 +109,14 @@ namespace CGAL
             it.cont(); ++it )
       {
         to_erase.push_back(it);
-        if ( !amap.template is_free<i+1>(it) && dg1==NULL )
+        if ( !amap.template is_free<i+1>(it) && dg1==amap.null_handle )
         { dg1=it; dg2=amap.template beta<i+1>(it); }
         amap.mark(it, mark);
         ++res;
       }
 
       // We group the two (i+1)-cells incident if they exist.
-      if ( dg1!=NULL )
+      if ( dg1!=amap.null_handle )
         CGAL::internal::Group_attribute_functor_run<CMap, i+1>::
             run(&amap, dg1, dg2);
 
@@ -334,7 +334,7 @@ namespace CGAL
       size_t res = 0;
 
       typename CMap::Dart_handle d1, d2;
-      typename CMap::Dart_handle dg1=NULL, dg2=NULL;
+      typename CMap::Dart_handle dg1=amap.null_handle, dg2=amap.null_handle;
 
       int mark = amap.get_new_mark();
 //      int mark_modified_darts = amap.get_new_mark();
@@ -345,14 +345,14 @@ namespace CGAL
             it.cont(); ++it )
       {
         to_erase.push_back(it);
-        if ( !amap.template is_free<0>(it) && dg1==NULL )
+        if ( !amap.template is_free<0>(it) && dg1==amap.null_handle )
         { dg1=it; dg2=amap.template beta<0>(it); }
         amap.mark(it, mark);
         ++res;
       }
 
       // We group the two edges incident if they exist.
-      if ( dg1!=NULL )
+      if ( dg1!=amap.null_handle )
         CGAL::internal::Group_attribute_functor_run<CMap, 1>::
             run(&amap, dg1, dg2);
 
@@ -507,7 +507,7 @@ namespace CGAL
       size_t res = 0;
 
       typename CMap::Dart_handle d1, d2;
-      typename CMap::Dart_handle dg1=NULL, dg2=NULL;
+      typename CMap::Dart_handle dg1=amap.null_handle, dg2=amap.null_handle;
 
       int mark = amap.get_new_mark();
       int mark_modified_darts = amap.get_new_mark();
@@ -520,14 +520,14 @@ namespace CGAL
             it.cont(); ++it )
       {
         to_erase.push_back(it);
-        if ( !amap.template is_free<i-1>(it) && dg1==NULL )
+        if ( !amap.template is_free<i-1>(it) && dg1==amap.null_handle )
         { dg1=it; dg2=amap.template beta<i-1>(it); }
         amap.mark(it, mark);
         ++res;
       }
 
       // We group the two (i+1)-cells incident if they exist.
-      if ( dg1!=NULL )
+      if ( dg1!=amap.null_handle )
          CGAL::internal::Group_attribute_functor_run<CMap,i-1>::
              run(&amap, dg1, dg2);
 
@@ -656,7 +656,7 @@ namespace CGAL
       size_t res = 0;
 
       typename CMap::Dart_handle d1, d2;
-      typename CMap::Dart_handle dg1=NULL, dg2=NULL;
+      typename CMap::Dart_handle dg1=amap.null_handle, dg2=amap.null_handle;
 
       int mark = amap.get_new_mark();
 //      int mark_modified_darts = amap.get_new_mark();
@@ -667,7 +667,7 @@ namespace CGAL
             it.cont(); ++it )
       {
         to_erase.push_back(it);
-        if ( dg1==NULL && !amap.template is_free<0>(it) &&
+        if ( dg1==amap.null_handle && !amap.template is_free<0>(it) &&
              !amap.template is_free<1>(it) )
         { dg1=amap.template beta<0>(it); dg2=amap.template beta<1>(it); }
         amap.mark(it, mark);
@@ -675,7 +675,7 @@ namespace CGAL
       }
 
       // We group the two vertices incident if they exist.
-      if ( dg1!=NULL )
+      if ( dg1!=amap.null_handle )
         CGAL::internal::Group_attribute_functor_run<CMap, 0, 1>::
             run(&amap, dg1, dg2);
 
diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_storages.h b/Combinatorial_map/include/CGAL/Combinatorial_map_storages.h
index 4ed0bb92a06..9edfb54c45d 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_storages.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_storages.h
@@ -58,6 +58,9 @@ namespace CGAL {
     typedef typename Dart_container::const_iterator Dart_const_handle;
     typedef typename Dart_container::size_type      size_type;
 
+    typedef CGAL::Void* Null_handle_type;
+    static Null_handle_type null_handle;
+
     typedef Items_ Items;
     typedef Alloc_ Alloc;
 
@@ -396,6 +399,11 @@ namespace CGAL {
     typename Helper::Attribute_containers mattribute_containers;
   };
 
+  /// null_handle
+  template < unsigned int d_, class Items_, class Alloc_ >
+  typename Combinatorial_map_storage_1<d_, Items_, Alloc_>::Null_handle_type
+  Combinatorial_map_storage_1<d_, Items_, Alloc_>::null_handle = NULL;
+
 #ifdef CGAL_CMAP_DEPRECATED
   /// Allocation of static data members
   /// mnull_dart_container
diff --git a/Combinatorial_map/include/CGAL/Dart.h b/Combinatorial_map/include/CGAL/Dart.h
index a69216995fa..fb420306afc 100644
--- a/Combinatorial_map/include/CGAL/Dart.h
+++ b/Combinatorial_map/include/CGAL/Dart.h
@@ -263,17 +263,6 @@ namespace CGAL {
         (mattribute_handles);
     }
 
-    /// Set the handle on the i th attribute
-    template<int i>
-    void set_attribute( typename Attribute_handle<i>::type ahandle )
-    {
-      CGAL_static_assertion_msg(Helper::template Dimension_index<i>::value>=0,
-                     "set_attribute<i> called but i-attributes are disabled.");
-      CGAL::cpp11::get<Helper::template Dimension_index<i>::value>
-        (mattribute_handles) = ahandle;
-      if (ahandle!=NULL) ahandle->inc_nb_refs();
-    }
-
     /** Return the mark value of a given mark number.
      * @param amark the mark number.
      * @return the value for this number.
diff --git a/Combinatorial_map/include/CGAL/Dart_iterators.h b/Combinatorial_map/include/CGAL/Dart_iterators.h
index 4f7fa122fe4..4385e7baa5d 100644
--- a/Combinatorial_map/include/CGAL/Dart_iterators.h
+++ b/Combinatorial_map/include/CGAL/Dart_iterators.h
@@ -166,7 +166,7 @@ namespace CGAL {
     Self& operator++()
     {
       CGAL_assertion(this->cont());
-      this->set_current_dart(NULL);
+      this->set_current_dart(this->mmap->null_handle);
       this->mprev_op = OP_END;
       return *this;
     }
@@ -248,7 +248,7 @@ namespace CGAL {
 
         if ((*this)==this->mfirst_dart)
         {
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
           this->mprev_op = OP_END;
         }
       }
@@ -256,7 +256,7 @@ namespace CGAL {
       {
         if (this->mmap->is_free(*this, 1))
         {
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
           this->mprev_op = OP_END;
         }
         else
@@ -338,7 +338,7 @@ namespace CGAL {
 
         if ((*this)==this->mfirst_dart)
         {
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
           this->mprev_op = OP_END;
         }
       }
@@ -346,7 +346,7 @@ namespace CGAL {
       {
         if (this->mmap->is_free(*this, 0))
         {
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
           this->mprev_op = OP_END;
         }
         else
@@ -402,7 +402,7 @@ namespace CGAL {
       CGAL_assertion(this->cont());
       if ((*this)!=this->mfirst_dart || this->mmap->is_free(*this, Bi))
       {
-        this->set_current_dart(NULL);
+        this->set_current_dart(this->mmap->null_handle);
         this->mprev_op = OP_END;
       }
       else
@@ -485,7 +485,7 @@ namespace CGAL {
           else
           {
             this->mprev_op = OP_END;
-            this->set_current_dart(NULL);
+            this->set_current_dart(this->mmap->null_handle);
           }
         }
       }
@@ -500,7 +500,7 @@ namespace CGAL {
         else
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
       }
       else if (mcurdart==2)
@@ -514,7 +514,7 @@ namespace CGAL {
       {
         CGAL_assertion (mcurdart==3);
         this->mprev_op = OP_END;
-        this->set_current_dart(NULL);
+        this->set_current_dart(this->mmap->null_handle);
       }
 
       return *this;
@@ -623,7 +623,7 @@ namespace CGAL {
       mexist_betaj(false),
       mprev_betaj(false),
       mfirst_border(true)
-    { if (adart!=NULL)
+    { if (adart!=this->mmap->null_handle)
         mexist_betaj=!this->mmap->is_free(adart, delta); }
     
     /// Main constructor.
@@ -634,7 +634,7 @@ namespace CGAL {
       mexist_betaj(false),
       mprev_betaj(false),
       mfirst_border(true)
-    { if (adart!=NULL)
+    { if (adart!=this->mmap->null_handle)
         mexist_betaj=!this->mmap->is_free(adart, delta); }
 
     /// Prefix ++ operator.
@@ -654,7 +654,7 @@ namespace CGAL {
         ++mit;
         this->mprev_op = mit.prev_operation();
         if ( !mit.cont() ) 
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         else
         {          
           if ( !mfirst_border ) 
@@ -723,7 +723,7 @@ namespace CGAL {
       mexist_betaj(false),
       mprev_betaj(false),
       mfirst_border(true)
-    { if (adart!=NULL)
+    { if (adart!=this->mmap->null_handle)
         mexist_betaj=!this->mmap->is_free(adart, 1+delta); }
     
     /// Main constructor.
@@ -734,7 +734,7 @@ namespace CGAL {
       mexist_betaj(false),
       mprev_betaj(false),
       mfirst_border(true)
-    { if (adart!=NULL)
+    { if (adart!=this->mmap->null_handle)
         mexist_betaj=!this->mmap->is_free(adart, 1+delta); }
 
     /// Prefix ++ operator.
@@ -754,7 +754,7 @@ namespace CGAL {
         ++mit;
         this->mprev_op = mit.prev_operation();
         if ( !mit.cont() ) 
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         else
         {          
           if ( !mfirst_border ) 
@@ -854,7 +854,7 @@ namespace CGAL {
             if (this->mmap->is_free(this->mfirst_dart, Bi+1))
             {
               this->mprev_op = OP_END;
-              this->set_current_dart(NULL);
+              this->set_current_dart(this->mmap->null_handle);
             }
             else
             {
@@ -877,7 +877,7 @@ namespace CGAL {
             if (this->mmap->is_free(this->mfirst_dart, Bi+1))
             {
               this->mprev_op = OP_END;
-              this->set_current_dart(NULL);
+              this->set_current_dart(this->mmap->null_handle);
             }
             else
             {
@@ -892,7 +892,7 @@ namespace CGAL {
             if ((*this)==this->mfirst_dart)
             {
               this->mprev_op = OP_END;
-              this->set_current_dart(NULL);
+              this->set_current_dart(this->mmap->null_handle);
             }
             else
             {
@@ -909,7 +909,7 @@ namespace CGAL {
           if (this->mmap->is_free(*this, Bi))
           {
             this->mprev_op = OP_END;
-            this->set_current_dart(NULL);
+            this->set_current_dart(this->mmap->null_handle);
           }
           else
           {
@@ -923,7 +923,7 @@ namespace CGAL {
           if (this->mmap->is_free(*this, Bi+1))
           {
             this->mprev_op = OP_END;
-            this->set_current_dart(NULL);
+            this->set_current_dart(this->mmap->null_handle);
           }
           else
           {
@@ -1137,7 +1137,7 @@ namespace CGAL {
       { this->mprev_op = OP_POP; }
       else
       {
-        this->set_current_dart(NULL);
+        this->set_current_dart(this->mmap->null_handle);
         this->mprev_op = OP_END;
       }
       return *this;
@@ -1176,7 +1176,7 @@ namespace CGAL {
       Base(amap, adart),
       mmark_number(amark)
     {
-      if (adart!=NULL)
+      if (adart!=this->mmap->null_handle)
       {
         this->mmap->mark_null_dart(mmark_number);
         this->mmap->mark(adart, mmark_number);
@@ -1198,13 +1198,13 @@ namespace CGAL {
     {
       CGAL_assertion(mmark_number != -1);
       CGAL_assertion(this->cont());
-      Dart_handle nd = NULL;
+      Dart_handle nd = this->mmap->null_handle;
       
       for ( unsigned int k=0; k<i; ++k )
       {
         if ( this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1221,7 +1221,7 @@ namespace CGAL {
       {
         if ( this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1235,7 +1235,7 @@ namespace CGAL {
         }
       }
 
-      if (nd == NULL)
+      if (nd == this->mmap->null_handle)
       {
         if (!mto_treat.empty())
         {
@@ -1246,7 +1246,7 @@ namespace CGAL {
         else
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
       }
       
@@ -1288,7 +1288,7 @@ namespace CGAL {
       Base(amap, adart),
       mmark_number(amark)
     {
-      if (adart!=NULL)
+      if (adart!=this->mmap->null_handle)
       {
         this->mmap->mark(adart, mmark_number);
         this->mmap->mark_null_dart(mmark_number);
@@ -1311,13 +1311,13 @@ namespace CGAL {
       CGAL_assertion(mmark_number != -1);
       CGAL_assertion(this->cont());
 
-      Dart_handle nd = NULL;
+      Dart_handle nd = this->mmap->null_handle;
       
       for ( unsigned int k=2; k<=d; ++k )
       {
         if ( this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1331,7 +1331,7 @@ namespace CGAL {
         }
       }
 
-      if (nd == NULL)
+      if (nd == this->mmap->null_handle)
       {
         if (!mto_treat.empty())
         {
@@ -1343,7 +1343,7 @@ namespace CGAL {
         else
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
       }
       
@@ -1384,7 +1384,7 @@ namespace CGAL {
                                      int amark):
       Base(amap, adart),
       mmark_number(amark)
-    { if (adart!=NULL)
+    { if (adart!=this->mmap->null_handle)
       {
         this->mmap->mark(adart, mmark_number);
         this->mmap->mark_null_dart(mmark_number);
@@ -1407,13 +1407,13 @@ namespace CGAL {
       CGAL_assertion(mmark_number != -1);
       CGAL_assertion(this->cont());
 
-      Dart_handle nd = NULL;
+      Dart_handle nd = this->mmap->null_handle;
 
       for ( unsigned int k=2; k<=d; ++k )
       {
         if ( this->is_unmarked2((*this), 0, k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, 0, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1427,7 +1427,7 @@ namespace CGAL {
         }
         if ( this->is_unmarked2((*this), k, 1, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k, 1);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1443,7 +1443,7 @@ namespace CGAL {
         {
           if ( this->is_unmarked2((*this), k, l, mmark_number) )
           {
-            if (nd == NULL)
+            if (nd == this->mmap->null_handle)
             {
               nd = this->mmap->beta(*this, k, l);
               CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1457,7 +1457,7 @@ namespace CGAL {
           }
           if ( this->is_unmarked2((*this), l, k, mmark_number) )
           {
-            if (nd == NULL)
+            if (nd == this->mmap->null_handle)
             {
               nd = this->mmap->beta(*this, l, k);
               CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1472,7 +1472,7 @@ namespace CGAL {
         }
       }
 
-      if (nd == NULL)
+      if (nd == this->mmap->null_handle)
       {
         if (!mto_treat.empty())
         {
@@ -1484,7 +1484,7 @@ namespace CGAL {
         else
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
       }
       
@@ -1720,7 +1720,7 @@ namespace CGAL {
           if ((*this)==this->mmap->null_dart_handle)
           {
             this->mprev_op = OP_END;
-            this->set_current_dart(NULL);
+            this->set_current_dart(this->mmap->null_handle);
           }
           else
           {
@@ -1732,7 +1732,7 @@ namespace CGAL {
           if ((*this)==this->mfirst_dart)
           {
             this->mprev_op = OP_END;
-            this->set_current_dart(NULL);
+            this->set_current_dart(this->mmap->null_handle);
           }
           else
             this->mprev_op = OP_BETA0I;
@@ -1744,7 +1744,7 @@ namespace CGAL {
         if ((*this) == this->mmap->null_dart_handle)
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
         else
           this->mprev_op = OP_BETA21;
@@ -1904,7 +1904,7 @@ namespace CGAL {
     {
       CGAL_assertion( d>=3 && d<=Map::dimension );
       CGAL_assertion( i>=3 && i<=Map::dimension );
-      if (adart!=NULL)
+      if (adart!=this->mmap->null_handle)
       {
         this->mmap->mark(adart, mmark_number);
         this->mmap->mark_null_dart(mmark_number);
@@ -1927,13 +1927,13 @@ namespace CGAL {
       CGAL_assertion(mmark_number != -1);
       CGAL_assertion(this->cont());
 
-      Dart_handle nd = NULL;
+      Dart_handle nd = this->mmap->null_handle;
 
       for ( int k=0; k<2; ++k )
       {
         if ( this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1952,7 +1952,7 @@ namespace CGAL {
         if ( k!=i-1 && k!=i && k!=i+1 && 
              this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -1966,7 +1966,7 @@ namespace CGAL {
         }
       }
 
-      if (nd == NULL)
+      if (nd == this->mmap->null_handle)
       {
         if (!mto_treat.empty())
         {
@@ -1977,7 +1977,7 @@ namespace CGAL {
         else
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
       }
       
@@ -2026,7 +2026,7 @@ namespace CGAL {
       mmark_number(amark)
     {
       CGAL_assertion( i>=3 && i<=Map::dimension );
-      if (adart!=NULL)
+      if (adart!=this->mmap->null_handle)
       {
         this->mmap->mark(adart, mmark_number);
         this->mmap->mark_null_dart(mmark_number);
@@ -2049,13 +2049,13 @@ namespace CGAL {
       CGAL_assertion(mmark_number != -1);
       CGAL_assertion(this->cont());
 
-      Dart_handle nd = NULL;
+      Dart_handle nd = this->mmap->null_handle;
 
       for ( int k=1; k>=0; --k )
       {
         if ( this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -2073,7 +2073,7 @@ namespace CGAL {
         if ( k!=i-1 && k!=i && k!=i+1 && 
              this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -2087,7 +2087,7 @@ namespace CGAL {
         }
       }
 
-      if (nd == NULL)
+      if (nd == this->mmap->null_handle)
       {
         if (!mto_treat.empty())
         {
@@ -2098,7 +2098,7 @@ namespace CGAL {
         else
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
       }
       
@@ -2143,7 +2143,7 @@ namespace CGAL {
                                            int amark):
       Base(amap, adart),
       mmark_number(amark)
-    { if (adart!=NULL)
+    { if (adart!=this->mmap->null_handle)
       {
         this->mmap->mark(adart, mmark_number);
         this->mmap->mark_null_dart(mmark_number);
@@ -2166,13 +2166,13 @@ namespace CGAL {
       CGAL_assertion(mmark_number != -1);
       CGAL_assertion(this->cont());
 
-      Dart_handle nd = NULL;
+      Dart_handle nd = this->mmap->null_handle;
 
       for ( unsigned int k=3; k<=d; ++k )
       {
         if ( this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -2186,7 +2186,7 @@ namespace CGAL {
         }
       }
 
-      if (nd == NULL)
+      if (nd == this->mmap->null_handle)
       {
         if (!mto_treat.empty())
         {
@@ -2197,7 +2197,7 @@ namespace CGAL {
         else
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
       }
       
@@ -2269,7 +2269,7 @@ namespace CGAL {
                                            int amark):
       Base(amap, adart),
       mmark_number(amark)
-    { if ( adart!=NULL)
+    { if ( adart!=this->mmap->null_handle)
       {
         this->mmap->mark(adart, mmark_number);
         this->mmap->mark_null_dart(mmark_number);
@@ -2292,13 +2292,13 @@ namespace CGAL {
       CGAL_assertion(mmark_number != -1);
       CGAL_assertion(this->cont());
 
-      Dart_handle nd = NULL;
+      Dart_handle nd = this->mmap->null_handle;
 
       for ( unsigned int k=4; k<=d; ++k )
       {
         if ( this->is_unmarked((*this), k, mmark_number) )
         {
-          if (nd == NULL)
+          if (nd == this->mmap->null_handle)
           {
             nd = this->mmap->beta(*this, k);
             CGAL_assertion(nd!=this->mmap->null_dart_handle);
@@ -2312,7 +2312,7 @@ namespace CGAL {
         }
       }
 
-      if (nd == NULL)
+      if (nd == this->mmap->null_handle)
       {
         if (!mto_treat.empty())
         {
@@ -2323,7 +2323,7 @@ namespace CGAL {
         else
         {
           this->mprev_op = OP_END;
-          this->set_current_dart(NULL);
+          this->set_current_dart(this->mmap->null_handle);
         }
       }
       
diff --git a/Combinatorial_map/test/Combinatorial_map/Combinatorial_map_copy_test.cpp b/Combinatorial_map/test/Combinatorial_map/Combinatorial_map_copy_test.cpp
index bd14a5f05d2..69f1dd725a2 100644
--- a/Combinatorial_map/test/Combinatorial_map/Combinatorial_map_copy_test.cpp
+++ b/Combinatorial_map/test/Combinatorial_map/Combinatorial_map_copy_test.cpp
@@ -167,7 +167,7 @@ struct CreateAttributes
     for(typename Map::Dart_range::iterator it=map.darts().begin(),
         itend=map.darts().end(); it!=itend; ++it)
     {
-      if ( it->template attribute<i>()==NULL )
+      if ( it->template attribute<i>()==map.null_handle )
         map.template set_attribute<i>
             (it, map.template create_attribute<i>(++nb));
     }
diff --git a/Linear_cell_complex/demo/Linear_cell_complex/Linear_cell_complex_3_subdivision.cpp b/Linear_cell_complex/demo/Linear_cell_complex/Linear_cell_complex_3_subdivision.cpp
index 18d37179120..74667e0e9de 100644
--- a/Linear_cell_complex/demo/Linear_cell_complex/Linear_cell_complex_3_subdivision.cpp
+++ b/Linear_cell_complex/demo/Linear_cell_complex/Linear_cell_complex_3_subdivision.cpp
@@ -37,7 +37,6 @@ public:
   Vertex operator  () (Vertex & v) const
   {
     Dart_handle d = v.dart ();
-    CGAL_assertion (d != NULL);
 
     int degree = 0;
     bool open = false;
@@ -83,7 +82,7 @@ flip_edge (LCC & m, Dart_handle d)
   CGAL_assertion ( !m.is_free(d,1) && !m.is_free(d,0) );
   CGAL_assertion ( !m.is_free(m.beta(d,2), 0) && !m.is_free(m.beta(d, 2), 1) );
   
-  if (!CGAL::is_removable<LCC,1>(m,d)) return NULL;
+  if (!CGAL::is_removable<LCC,1>(m,d)) return LCC::null_handle;
 
   Dart_handle d1 = m.beta(d,1);
   Dart_handle d2 = m.beta(d,2,0);
@@ -170,12 +169,11 @@ subdivide_lcc_3 (LCC & m)
 
   // 4) We flip all the old edges.
   m.negate_mark (mark);  // Now only new darts are marked.
-  Dart_handle d2 = NULL;
+  Dart_handle d2 =LCC::null_handle;
   for (LCC::Dart_range::iterator it (m.darts().begin ());
        it != m.darts().end ();)
   {
     d2 = it++;
-    CGAL_assertion (d2 != NULL);
     if (!m.is_marked (d2, mark))  // This is an old dart.
     {
       // We process only the last dart of a same edge.
diff --git a/Linear_cell_complex/demo/Linear_cell_complex/Linear_cell_complex_pqq_subdivision.cpp b/Linear_cell_complex/demo/Linear_cell_complex/Linear_cell_complex_pqq_subdivision.cpp
index fda845861d7..bb2f1d7a2f3 100644
--- a/Linear_cell_complex/demo/Linear_cell_complex/Linear_cell_complex_pqq_subdivision.cpp
+++ b/Linear_cell_complex/demo/Linear_cell_complex/Linear_cell_complex_pqq_subdivision.cpp
@@ -37,7 +37,6 @@ public:
   Vertex operator  () (Vertex & v) const
   {
     Dart_handle d = v.dart ();
-    CGAL_assertion (d != NULL);
 
     // Old points aren't concerned.
     if (mlcc.is_marked(d, old))
@@ -119,7 +118,6 @@ public:
   Vertex operator  () (Vertex & v) const
   {
     Dart_handle d = v.dart ();
-    CGAL_assertion (d != NULL);
 
     // Just old points are concerned.
     if (!mlcc.is_marked(d, old))
diff --git a/Linear_cell_complex/demo/Linear_cell_complex/MainWindow.cpp b/Linear_cell_complex/demo/Linear_cell_complex/MainWindow.cpp
index 2dfc3cd28e6..5d906b17f2b 100644
--- a/Linear_cell_complex/demo/Linear_cell_complex/MainWindow.cpp
+++ b/Linear_cell_complex/demo/Linear_cell_complex/MainWindow.cpp
@@ -167,7 +167,7 @@ void MainWindow::clear_all()
 
 void MainWindow::on_new_volume(Dart_handle adart)
 {
-  CGAL_assertion( scene.lcc->attribute<3>(adart)==NULL);
+  CGAL_assertion( scene.lcc->attribute<3>(adart)==LCC::null_handle);
   CGAL::Set_i_attribute_functor<LCC, 3>::
       run(scene.lcc, adart, scene.lcc->create_attribute<3>());
   update_volume_list_add(scene.lcc->attribute<3>(adart));
@@ -178,7 +178,7 @@ void MainWindow::init_all_new_volumes()
   for (LCC::One_dart_per_cell_range<3>::iterator
        it(scene.lcc->one_dart_per_cell<3>().begin());
        it.cont(); ++it)
-    if ( scene.lcc->attribute<3>(it)==NULL )
+    if ( scene.lcc->attribute<3>(it)==LCC::null_handle )
     { on_new_volume(it); }
 }
 
@@ -751,7 +751,7 @@ void MainWindow::on_actionMerge_all_volumes_triggered()
   timer.start();
 #endif
 
-  Dart_handle prev = NULL;
+  Dart_handle prev = scene.lcc->null_handle;
   for (LCC::Dart_range::iterator it(scene.lcc->darts().begin()),
        itend=scene.lcc->darts().end(); it!=itend; )
   {
@@ -762,7 +762,7 @@ void MainWindow::on_actionMerge_all_volumes_triggered()
     {
       CGAL::remove_cell<LCC,2>(*scene.lcc,it);
       itend=scene.lcc->darts().end();
-      if ( prev==NULL ) it=scene.lcc->darts().begin();
+      if ( prev==scene.lcc->null_handle ) it=scene.lcc->darts().begin();
       else { it=prev; if ( it!=itend ) ++it; }
     }
     else
diff --git a/Linear_cell_complex/demo/Linear_cell_complex/Viewer.cpp b/Linear_cell_complex/demo/Linear_cell_complex/Viewer.cpp
index 63e7a05bf84..2562a694b52 100644
--- a/Linear_cell_complex/demo/Linear_cell_complex/Viewer.cpp
+++ b/Linear_cell_complex/demo/Linear_cell_complex/Viewer.cpp
@@ -76,7 +76,7 @@ void Viewer::drawFacet(Dart_const_handle ADart)
 {
   LCC &m = *scene->lcc;
   ::glBegin(GL_POLYGON);
-  CGAL_assertion( m.attribute<3>(ADart)!=NULL );
+  CGAL_assertion( m.attribute<3>(ADart)!=LCC::null_handle );
 
   //  double r = (double)ADart->attribute<3>()->info().r()/255.0;
   double r = (double)m.info<3>(ADart).color().r()/255.0;
@@ -128,7 +128,7 @@ void Viewer::drawEdges(Dart_const_handle ADart)
   {
     LCC::Point p = m.point(it);
     Dart_const_handle d2 = m.other_extremity(it);
-    if ( d2!=NULL )
+    if ( d2!=LCC::null_handle )
     {
       LCC::Point p2 = m.point(d2);
       glVertex3f( p.x(),p.y(),p.z());
@@ -158,7 +158,7 @@ void Viewer::draw_one_vol(Dart_const_handle adart, bool filled)
     for (LCC::One_dart_per_incident_cell_range<1,3>::const_iterator
            it(m,adart); it.cont(); ++it)
     {
-      if ( m.other_extremity(it)!=NULL )
+      if ( m.other_extremity(it)!=LCC::null_handle )
       {
         LCC::Point p1 = m.point(it);
         LCC::Point p2 = m.point(m.other_extremity(it));
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex.h
index 2a1b76a674b..dc4ba4aebae 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex.h
@@ -321,7 +321,7 @@ namespace CGAL {
       for (typename Dart_range::const_iterator it(this->darts().begin()),
              itend(this->darts().end()); valid && it != itend; ++it)
       {
-        if ( vertex_attribute(it) == NULL )
+        if ( vertex_attribute(it)==null_handle )
         {
           std::cerr << "Map not valid: dart "<<&(*it)
                     <<" does not have a vertex."<< std::endl;
@@ -344,7 +344,7 @@ namespace CGAL {
       bool samegeometry = true;
       for ( ; samegeometry && it1.cont() && it2.cont(); ++it1, ++it2)
       {
-        if ( this->other_extremity(it2)!=NULL &&
+        if ( this->other_extremity(it2)!=null_handle &&
              point(it1)!=point(this->other_extremity(it2)) )
           samegeometry = false;
       }
@@ -686,7 +686,7 @@ namespace CGAL {
 
       Dart_handle first = CGAL::insert_cell_0_in_cell_2(*this, dh, v);
 
-      if ( first== NULL ) // If the triangulated facet was made of one dart
+      if ( first==null_handle ) // If the triangulated facet was made of one dart
         erase_vertex_attribute(v);
 
 #ifdef CGAL_CMAP_TEST_VALID_INSERTIONS
@@ -739,7 +739,7 @@ namespace CGAL {
      * simultaneously through all the darts of the two lcc and we have
      * each time of the iteration two "dual" darts.
      */
-    Dart_handle dual_points_at_barycenter(Self & alcc, Dart_handle adart=NULL)
+    Dart_handle dual_points_at_barycenter(Self & alcc, Dart_handle adart=null_handle)
     {
       Dart_handle res = Base::dual(alcc, adart);
 
@@ -750,7 +750,7 @@ namespace CGAL {
       for (typename Dart_range::iterator it(this->darts().begin());
            it!=this->darts().end(); ++it, ++it2)
       {
-        if (vertex_attribute(it2) == NULL)
+        if (vertex_attribute(it2)==null_handle)
         {
           alcc.set_vertex_attribute(it2, alcc.create_vertex_attribute
                                     (barycenter<dimension>(it)));
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_constructors.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_constructors.h
index 01a8951f176..f828f59596a 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_constructors.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_constructors.h
@@ -63,7 +63,7 @@ namespace CGAL {
 
     std::string txt;
     typename LCC::FT x, y;
-    Dart_handle d1 = NULL, d2 = NULL;
+    Dart_handle d1 = alcc.null_handle, d2 = alcc.null_handle;
     unsigned int v1, v2;
   
     unsigned int nbSommets = 0;
@@ -76,7 +76,7 @@ namespace CGAL {
       {
         std::cout << "Problem: file does not contain enough vertices."
                   << std::endl;
-        return NULL;
+        return alcc.null_handle;
       }
 
       ais >> x >> y;
@@ -92,7 +92,7 @@ namespace CGAL {
       {
         std::cout << "Problem: file does not contain enough edges."
                   << std::endl;
-        return NULL;
+        return alcc.null_handle;
       }
 
       // We read an egde (given by the number of its two vertices).
@@ -115,8 +115,8 @@ namespace CGAL {
     List_iterator it;
     LCC_iterator  it2;
 
-    Dart_handle first = NULL;
-    Dart_handle prec = NULL;
+    Dart_handle first = alcc.null_handle;
+    Dart_handle prec = alcc.null_handle;
     typename LCC::Point sommet1, sommet2;
   
     for (unsigned int i = 0; i < initVertices.size(); ++i)
@@ -182,10 +182,10 @@ namespace CGAL {
     CGAL_static_assertion( LCC::dimension>=2 && LCC::ambient_dimension==2 );
     
     // Case of empty triangulations.
-    if (atr.number_of_vertices() == 0) return NULL;
+    if (atr.number_of_vertices() == 0) return LCC::null_handle;
 
     // Check the dimension.
-    if (atr.dimension() != 2) return NULL;
+    if (atr.dimension() != 2) return LCC::null_handle;
     CGAL_assertion(atr.is_valid());
 
     typedef typename Triangulation::Vertex_handle         TVertex_handle;
@@ -213,8 +213,8 @@ namespace CGAL {
     
     itmap_tcell maptcell_it;
 
-    typename LCC::Dart_handle res=NULL, dart=NULL;
-    typename LCC::Dart_handle cur=NULL, neighbor=NULL;
+    typename LCC::Dart_handle res=LCC::null_handle, dart=LCC::null_handle;
+    typename LCC::Dart_handle cur=LCC::null_handle, neighbor=LCC::null_handle;
 
     for (it = atr.all_faces_begin(); it != atr.all_faces_end(); ++it)
     {
@@ -228,7 +228,7 @@ namespace CGAL {
                                  TV[it->vertex(1)],
                                  TV[it->vertex(2)]);
 
-        if ( dart==NULL )
+        if ( dart==LCC::null_handle )
         {
           if ( it->vertex(0) == atr.infinite_vertex() )
             dart = res;
@@ -265,7 +265,7 @@ namespace CGAL {
       }
     }
 
-    CGAL_assertion(dart!=NULL);
+    CGAL_assertion(dart!=LCC::null_handle);
     return dart;
   }
   
@@ -285,10 +285,10 @@ namespace CGAL {
     CGAL_static_assertion( LCC::dimension>=3 && LCC::ambient_dimension==3 );
     
     // Case of empty triangulations.
-    if (atr.number_of_vertices() == 0) return NULL;
+    if (atr.number_of_vertices() == 0) return LCC::null_handle;
 
     // Check the dimension.
-    if (atr.dimension() != 3) return NULL;
+    if (atr.dimension() != 3) return LCC::null_handle;
     CGAL_assertion(atr.is_valid());
 
     typedef typename Triangulation::Vertex_handle    TVertex_handle;
@@ -316,8 +316,8 @@ namespace CGAL {
     
     itmap_tcell maptcell_it;
 
-    typename LCC::Dart_handle res=NULL, dart=NULL;
-    typename LCC::Dart_handle cur=NULL, neighbor=NULL;
+    typename LCC::Dart_handle res=LCC::null_handle, dart=LCC::null_handle;
+    typename LCC::Dart_handle cur=LCC::null_handle, neighbor=LCC::null_handle;
 
     for (it = atr.cells_begin(); it != atr.cells_end(); ++it)
     {
@@ -332,7 +332,7 @@ namespace CGAL {
                                     TV[it->vertex(2)],
                                     TV[it->vertex(3)]);
 
-        if ( dart==NULL )
+        if ( dart==LCC::null_handle )
         {
           if ( it->vertex(0) == atr.infinite_vertex() )
             dart = res;
@@ -375,7 +375,7 @@ namespace CGAL {
         (*mytc)[it] = res;
       }
     }
-    CGAL_assertion(dart!=NULL);
+    CGAL_assertion(dart!=LCC::null_handle);
     return dart;
   }
 
@@ -401,20 +401,20 @@ namespace CGAL {
     Halfedge_handle_map TC;
 
     itmap_hds it;
-    typename LCC::Dart_handle d = NULL, prev = NULL;
-    typename LCC::Dart_handle firstFacet = NULL, firstAll = NULL;
+    typename LCC::Dart_handle d = LCC::null_handle, prev = LCC::null_handle;
+    typename LCC::Dart_handle firstFacet = LCC::null_handle, firstAll = LCC::null_handle;
 
     // First traversal to build the darts and link them.
     for (Facet_iterator i = apoly.facets_begin(); i != apoly.facets_end(); ++i)
     {
       HF_circulator j = i->facet_begin();
-      prev = NULL;
+      prev = LCC::null_handle;
       do
       {
         d = alcc.create_dart();
         TC[j] = d;
       
-        if (prev != NULL) alcc.template link_beta<1>(prev, d);
+        if (prev != LCC::null_handle) alcc.template link_beta<1>(prev, d);
         else firstFacet = d;
         it = TC.find(j->opposite());
         if (it != TC.end())
@@ -423,7 +423,7 @@ namespace CGAL {
       }
       while (++j != i->facet_begin());
       alcc.template link_beta<1>(prev, firstFacet);
-      if (firstAll == NULL) firstAll = firstFacet;
+      if (firstAll == LCC::null_handle) firstAll = firstFacet;
     }
 
     // Second traversal to update the geometry.
@@ -434,7 +434,7 @@ namespace CGAL {
       do
       {
         d = TC[j]; // Get the dart associated to the Halfedge
-        if (alcc.temp_vertex_attribute(d) == NULL)
+        if (alcc.temp_vertex_attribute(d) == LCC::null_handle)
         {
           alcc.set_vertex_attribute
             (d, alcc.create_vertex_attribute(j->opposite()->vertex()->point()));
@@ -537,7 +537,7 @@ namespace CGAL {
     if (!ais.good())
     {
       std::cout << "Error reading flux." << std::endl;
-      return NULL;
+      return LCC::null_handle;
     }
     CGAL::Polyhedron_3<typename LCC::Traits> P;
     ais >> P;
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_incremental_builder.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_incremental_builder.h
index 7f77f8308f8..937cf3e45e1 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_incremental_builder.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_incremental_builder.h
@@ -50,7 +50,7 @@ namespace CGAL {
     
     void begin_facet()
     {
-      CGAL_assertion( first_dart==NULL && prev_dart==NULL );
+      CGAL_assertion( first_dart==lcc.null_handle && prev_dart==lcc.null_handle );
       // std::cout<<"Begin facet: "<<std::flush;
     }
 
@@ -60,12 +60,12 @@ namespace CGAL {
       // std::cout<<i<<"  "<<std::flush;
       Dart_handle cur = lcc.create_dart(vertex_map[i]);
 
-      if ( prev_dart!=NULL )
+      if ( prev_dart!=lcc.null_handle )
       {
         lcc.template link_beta<1>(prev_dart, cur);
 
         Dart_handle opposite=find_dart_between(i,lcc.temp_vertex_attribute(prev_dart));
-        if ( opposite!=NULL )
+        if ( opposite!=lcc.null_handle )
         {
           CGAL_assertion( lcc.template is_free<2>(opposite) );
           lcc.template link_beta<2>(prev_dart, opposite);
@@ -85,12 +85,12 @@ namespace CGAL {
 
     void end_facet()
     {
-      CGAL_assertion( first_dart!=NULL && prev_dart!=NULL );
+      CGAL_assertion( first_dart!=lcc.null_handle && prev_dart!=lcc.null_handle );
       lcc.template link_beta<1>(prev_dart, first_dart);
 
       Dart_handle opposite =
         find_dart_between(first_vertex,lcc.temp_vertex_attribute(prev_dart));
-      if ( opposite!=NULL )
+      if ( opposite!=lcc.null_handle )
       {
         CGAL_assertion( lcc.template is_free<2>(opposite) );
         lcc.template link_beta<2>(prev_dart, opposite);
@@ -98,16 +98,16 @@ namespace CGAL {
 
       add_dart_in_vertex_to_dart_map( prev_dart, prev_vertex );
 
-      first_dart = NULL;
-      prev_dart = NULL;
+      first_dart = lcc.null_handle;
+      prev_dart = lcc.null_handle;
       // std::cout<<"  end facet."<<std::endl;
     }
 
     void begin_surface( std::size_t v, std::size_t /*f*/, std::size_t /*h*/)
     {
       new_vertices  = 0;
-      first_dart    = NULL;
-      prev_dart     = NULL;
+      first_dart    = lcc.null_handle;
+      prev_dart     = lcc.null_handle;
       vertex_map.clear();
       vertex_to_dart_map.clear();
       vertex_map.reserve(v);
@@ -131,12 +131,12 @@ namespace CGAL {
       {
         if ( lcc.temp_vertex_attribute(lcc.template beta<1>(*it))==vh ) return (*it);
       }
-      return NULL;
+      return lcc.null_handle;
     }
 
     void add_dart_in_vertex_to_dart_map( Dart_handle adart, size_type i )
     {
-      CGAL_assertion( adart!=NULL );
+      CGAL_assertion( adart!=lcc.null_handle );
       CGAL_assertion( !lcc.template is_free<1>(adart) );
       vertex_to_dart_map[i].push_back(adart);
     }
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_operations.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_operations.h
index d90687dd70c..3a212fbf1dc 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_operations.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_operations.h
@@ -56,7 +56,7 @@ namespace CGAL {
       start = amap.template beta<0>(start);
 
     if ( amap.template is_free<1>(start) ||
-         amap.other_extremity(amap.template beta<1>(start))==NULL )
+         amap.other_extremity(amap.template beta<1>(start))==LCC::null_handle )
       return normal;
 
     unsigned int nb = 0;
@@ -64,7 +64,7 @@ namespace CGAL {
   
     const Point* prev = &amap.point(start);
     const Point* curr = &amap.point(adart);
-    for ( ; adart!=start && amap.other_extremity(adart)!=NULL;
+    for ( ; adart!=start && amap.other_extremity(adart)!=LCC::null_handle;
           adart=amap.template beta<1>(adart) )
     {
       const Point* next = &amap.point( amap.other_extremity(adart));
diff --git a/Linear_cell_complex/include/CGAL/Linear_cell_complex_storages.h b/Linear_cell_complex/include/CGAL/Linear_cell_complex_storages.h
index c2988416b48..63b43d8fed0 100644
--- a/Linear_cell_complex/include/CGAL/Linear_cell_complex_storages.h
+++ b/Linear_cell_complex/include/CGAL/Linear_cell_complex_storages.h
@@ -58,6 +58,9 @@ namespace CGAL {
     typedef typename Dart_container::const_iterator Dart_const_handle;
     typedef typename Dart_container::size_type      size_type;
 
+    typedef CGAL::Void* Null_handle_type;
+    static Null_handle_type null_handle;
+
     typedef Items_ Items;
     typedef Alloc_ Alloc;
 
@@ -410,17 +413,31 @@ namespace CGAL {
     typename Helper::Attribute_containers mattribute_containers;
   };
 
+  /// null_handle
+  template <unsigned int d_, unsigned int ambient_dim,
+           class Traits_, class Items_, class Alloc_ >
+  typename Linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
+                                         Items_, Alloc_>::Null_handle_type
+  Linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
+                                Items_, Alloc_>::null_handle = NULL;
+
 #ifdef CGAL_CMAP_DEPRECATED
   /// Allocation of static data members
   /// mnull_dart_container
-  template<unsigned int d_, class Items_, class Alloc_ >
-  typename Linear_cell_complex_storage_1<d_, Items_, Alloc_>::Dart_container
-  Linear_cell_complex_storage_1<d_, Items_, Alloc_>::mnull_dart_container;
+  template <unsigned int d_, unsigned int ambient_dim,
+           class Traits_, class Items_, class Alloc_ >
+  typename Linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
+                                         Items_, Alloc_>::Dart_container
+  Linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
+                                         Items_, Alloc_>::mnull_dart_container;
 
   /// null_dart_handle
-  template < unsigned int d_, class Items_, class Alloc_ >
-  typename Linear_cell_complex_storage_1<d_, Items_, Alloc_>::Dart_handle
-  Linear_cell_complex_storage_1<d_, Items_, Alloc_>::null_dart_handle;
+  template <unsigned int d_, unsigned int ambient_dim,
+           class Traits_, class Items_, class Alloc_ >
+  typename Linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
+                                         Items_, Alloc_>::Dart_handle
+  Linear_cell_complex_storage_1<d_, ambient_dim, Traits_,
+                                         Items_, Alloc_>::null_dart_handle;
   // =  mnull_dart_container.emplace( std::bitset<NB_MARKS>() );
   // Does not work on windows => segfault
   // Thus we initialize null_dart_handle in the Combinatorial_map constructor
diff --git a/Linear_cell_complex/test/Linear_cell_complex/Linear_cell_complex_copy_test.cpp b/Linear_cell_complex/test/Linear_cell_complex/Linear_cell_complex_copy_test.cpp
index 403a64e8f9f..aa164791081 100644
--- a/Linear_cell_complex/test/Linear_cell_complex/Linear_cell_complex_copy_test.cpp
+++ b/Linear_cell_complex/test/Linear_cell_complex/Linear_cell_complex_copy_test.cpp
@@ -166,10 +166,10 @@ struct Converter_map9_points_into_map5_points
   (const Map9& map1, Map5& map2, Map9::Dart_const_handle dh1,
    Map5::Dart_handle dh2) const
   {
-    CGAL_assertion( map1.attribute<0>(dh1)!=NULL );
+    CGAL_assertion( map1.attribute<0>(dh1)!=map1.null_handle );
 
     Map5::Attribute_handle<0>::type res = map2.attribute<0>(dh2);
-    if ( res==NULL )
+    if ( res==map2.null_handle )
     {
       res = map2.create_attribute<0>();
     }
@@ -221,7 +221,7 @@ struct CreateAttributes
     for(typename Map::Dart_range::iterator it=map.darts().begin(),
         itend=map.darts().end(); it!=itend; ++it)
     {
-      if ( map.template attribute<i>(it)==NULL )
+      if ( map.template attribute<i>(it)==map.null_handle )
       {
         map.template set_attribute<i>(it, map.template create_attribute<i>());
         SetInfoIfNonVoid<Map, i>::run(map, map.template attribute<i>(it), ++nb);
diff --git a/Polyhedron/demo/Polyhedron/Scene_combinatorial_map_item.h b/Polyhedron/demo/Polyhedron/Scene_combinatorial_map_item.h
index 00a8b143eb4..566ce79f454 100644
--- a/Polyhedron/demo/Polyhedron/Scene_combinatorial_map_item.h
+++ b/Polyhedron/demo/Polyhedron/Scene_combinatorial_map_item.h
@@ -1,6 +1,7 @@
 #ifndef SCENE_COMBINATORIAL_MAP_ITEM_H
 #define SCENE_COMBINATORIAL_MAP_ITEM_H
 
+#define CGAL_CMAP_DEPRECATED 1
 
 //=========
 #include <CGAL/internal/corefinement/Combinatorial_map_for_corefinement.h>