diff --git a/Packages/Triangulation_2/include/CGAL/Constrained_Delaunay_triangulation_2.h b/Packages/Triangulation_2/include/CGAL/Constrained_Delaunay_triangulation_2.h
index 00008b89f2f..87aa355c731 100644
--- a/Packages/Triangulation_2/include/CGAL/Constrained_Delaunay_triangulation_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Constrained_Delaunay_triangulation_2.h
@@ -104,7 +104,7 @@ public:
   bool test_conflict(Face_handle fh, const Point& p) const; //deprecated
   bool test_conflict(const Point& p, Face_handle fh) const;
   void find_conflicts(const Point& p, std::list<Edge>& le,  //deprecated
-		      Face_handle hint= Face_handle(NULL)) const;
+		      Face_handle hint= Face_handle()) const;
   //  //template member functions, declared and defined at the end 
   // template <class OutputItFaces, class OutputItBoundaryEdges> 
   // std::pair<OutputItFaces,OutputItBoundaryEdges>
@@ -125,7 +125,7 @@ public:
    
 
   // INSERTION-REMOVAL
-  Vertex_handle insert(const Point & a, Face_handle start = Face_handle(NULL));
+  Vertex_handle insert(const Point & a, Face_handle start = Face_handle());
   Vertex_handle insert(const Point& p,
 		       Locate_type lt,
 		       Face_handle loc, int li );
@@ -146,7 +146,7 @@ public:
  
 protected:
   virtual Vertex_handle virtual_insert(const Point & a, 
-				       Face_handle start = Face_handle(NULL));
+				       Face_handle start = Face_handle());
   virtual Vertex_handle virtual_insert(const Point& a,
 				       Locate_type lt,
 				       Face_handle loc, 
@@ -183,7 +183,7 @@ public:
   get_conflicts_and_boundary(const Point  &p, 
 			     OutputItFaces fit, 
 			     OutputItBoundaryEdges eit,
-			     Face_handle start = Face_handle(NULL)) const {
+			     Face_handle start = Face_handle()) const {
     CGAL_triangulation_precondition( dimension() == 2);
     int li;
     Locate_type lt;
@@ -211,7 +211,7 @@ public:
   OutputItFaces
   get_conflicts (const Point  &p, 
 		 OutputItFaces fit, 
-		 Face_handle start= Face_handle(NULL)) const {
+		 Face_handle start= Face_handle()) const {
     std::pair<OutputItFaces,Emptyset_iterator> pp = 
       get_conflicts_and_boundary(p,fit,Emptyset_iterator(),start);
     return pp.first;
@@ -221,7 +221,7 @@ public:
   OutputItBoundaryEdges
   get_boundary_of_conflicts(const Point  &p, 
 			    OutputItBoundaryEdges eit, 
-			    Face_handle start= Face_handle(NULL)) const {
+			    Face_handle start= Face_handle()) const {
     std::pair<Emptyset_iterator, OutputItBoundaryEdges> pp = 
       get_conflicts_and_boundary(p,Emptyset_iterator(),eit,start);
     return pp.second;
@@ -546,7 +546,7 @@ remove(Vertex_handle v)
   // remove a vertex and updates the constrained edges of the new faces
   // precondition : there is no incident constraints
 {
-  CGAL_triangulation_precondition( v != NULL );
+  CGAL_triangulation_precondition( v != Vertex_handle() );
   CGAL_triangulation_precondition( ! is_infinite(v));
   CGAL_triangulation_precondition( ! are_there_incident_constraints(v));
   if  (dimension() <= 1)    Ctr::remove(v);
diff --git a/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_2.h b/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_2.h
index 63d676e7677..3bff2b8046f 100644
--- a/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_2.h
@@ -112,7 +112,7 @@ public:
 
   // INSERTION
   Vertex_handle insert(const Point& p, 
-			       Face_handle start = Face_handle(NULL) );
+			       Face_handle start = Face_handle() );
   Vertex_handle insert(const Point& p,
 		       Locate_type lt,
 		       Face_handle loc, 
@@ -161,7 +161,7 @@ public:
 
 protected:
   virtual Vertex_handle virtual_insert(const Point& a, 
-				       Face_handle start = Face_handle(NULL));
+				       Face_handle start = Face_handle());
   virtual Vertex_handle virtual_insert(const Point& a,
 				       Locate_type lt,
 				       Face_handle loc, 
@@ -577,7 +577,7 @@ intersect(Face_handle , int ,
 	    <<    std::endl
 	    << " intersecting constraints" << std::endl;
   CGAL_triangulation_assertion(false);
-  return Vertex_handle(NULL);
+  return Vertex_handle() ;
 }
 
 template <class Gt, class Tds, class Itag >
@@ -852,7 +852,7 @@ remove(Vertex_handle  v)
   // remove a vertex and updates the constrained edges of the new faces
   // precondition : there is no incident constraints
 {
-  CGAL_triangulation_precondition( v != NULL );
+  CGAL_triangulation_precondition( v != Vertex_handle() );
   CGAL_triangulation_precondition( ! is_infinite(v));
   CGAL_triangulation_precondition( ! are_there_incident_constraints(v));
     
@@ -987,7 +987,7 @@ triangulate_half_hole(List_edges & list_edges,  List_edges & new_edges)
       n1=(*current).first;
       ind1=(*current).second;
       // in case n1 is no longer a triangle of the new triangulation
-      if ( n1->neighbor(ind1) != NULL ) {
+      if ( n1->neighbor(ind1) != Face_handle() ) {
 	n=n1->neighbor(ind1);
 	//ind=n1->mirror_index(ind1); 
 	// mirror_index does not work in this case
@@ -998,7 +998,7 @@ triangulate_half_hole(List_edges & list_edges,  List_edges & new_edges)
       n2=(*next).first;
       ind2=(*next).second;
       // in case n2 is no longer a triangle of the new triangulation
-      if (n2->neighbor(ind2) != NULL ) {
+      if (n2->neighbor(ind2) != Face_handle() ) {
 	n=n2->neighbor(ind2); 
 	// ind=n2->mirror_index(ind2);
 	// mirror_index does not work in this case
diff --git a/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h b/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
index 10da4f06382..cc1df8c3d7e 100644
--- a/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_plus_2.h
@@ -115,7 +115,7 @@ public:
 
   // INSERTION
   Vertex_handle insert(const Point& a, 
-		       Face_handle start = Face_handle(NULL) );
+		       Face_handle start = Face_handle() );
   Vertex_handle insert(const Point& p,
 		       Locate_type lt,
 		       Face_handle loc, int li );
@@ -390,7 +390,7 @@ intersect(Face_handle , int ,
 	    <<    std::endl
 	    << " intersecting constraints" << std::endl;
   CGAL_triangulation_assertion(false);
-  return Vertex_handle(NULL);
+  return Vertex_handle();
 }
 
 template <class Tr>
diff --git a/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_sweep_2.h b/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_sweep_2.h
index 298597a77f7..6110d5dd79a 100644
--- a/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_sweep_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_sweep_2.h
@@ -156,8 +156,8 @@ public:
     bool is_removable(Face_handle fh)
       {
 	return ( (*fh).vertex(1) == (*fh).vertex(2) &&
-		 (*fh).neighbor(1) != NULL && 
-		 (*fh).neighbor(2) != NULL );
+		 (*fh).neighbor(1) != Face_handle() && 
+		 (*fh).neighbor(2) != Face_handle() );
       }
 
     void remove_flat(Face_handle fh) 
@@ -165,9 +165,9 @@ public:
 	CGAL_triangulation_precondition((*fh).vertex(1) == (*fh).vertex(2));
 	Face_handle f2= (*fh).neighbor(2);
 	Face_handle f1= (*fh).neighbor(1);
-	if ( f2 != NULL ) { (*f2).set_neighbor( (*f2).index(fh), f1);}
-	if ( f1 != NULL ) { (*f1).set_neighbor( (*f1).index(fh), f2);}
-	( (*fh). vertex(0))->set_face( f2!= NULL ? f2 : f1 );
+	if ( f2 != Face_handle() ) { (*f2).set_neighbor( (*f2).index(fh), f1);}
+	if ( f1 != Face_handle() ) { (*f1).set_neighbor( (*f1).index(fh), f2);}
+	( (*fh). vertex(0))->set_face( f2!= Face_handle() ? f2 : f1 );
 	_tr->delete_face(fh);
 	return;
       }
@@ -261,7 +261,7 @@ public:
       Neighbor_list down;
     
     public:
-      Chain(Ctriangulation* tr) : rm(NULL), up(tr), down(tr) {}
+      Chain(Ctriangulation* tr) : rm(), up(tr), down(tr) {}
       Vertex_handle right_most() { return rm;}
       Neighbor_list* up_list(){return &up;}
       Neighbor_list* down_list(){return &down;}
@@ -290,7 +290,7 @@ public:
  
  public:
     Constrained_triangulation_sweep_2()
-      : _tr(NULL), _lc(NULL), upper_chain(NULL)
+      : _tr(NULL), _lc(), upper_chain()
     {
     }
     
@@ -422,7 +422,7 @@ treat_in_edges(const Event_queue_iterator & event,
   if (loc == status.end()) { pch = &upper_chain;}
   else { pch = (Chain*)((*loc).second);}
   Vertex_handle w = pch->right_most();
-  if (w == NULL ) { // first event is treated
+  if (w == Vertex_handle() ) { // first event is treated
     pch->set_right_most(v);
     return v;
   }
@@ -455,11 +455,11 @@ treat_in_edges(const Event_queue_iterator & event,
   last= nl->up_visit(v,last);
 
   //delete flat newf if possible
-  // i. e. if at least one of its neighbor is not NULL
-  if ( newf->neighbor(2) != NULL  || newf->neighbor(1)!= NULL ) {
-    if (first == newf ) { // means newf->neighbor(1) == NULL
+  // i. e. if at least one of its neighbor is not Face_handle()
+  if ( newf->neighbor(2) != Face_handle()  || newf->neighbor(1)!= Face_handle() ) {
+    if (first == newf ) { // means newf->neighbor(1) == Face_handle()
        first  = newf->neighbor(2);}
-    if (last == newf) { // means newf->neighbor(2) == NULL
+    if (last == newf) { // means newf->neighbor(2) == Face_handle()
       last = newf->neighbor(1);}
     nl->remove_flat(newf);
   }
@@ -569,7 +569,7 @@ set_infinite_faces()
   Vertex_handle infinite= _tr->infinite_vertex();
 
   // Triangulation may be empty;
-  if (upper_chain.right_most() == NULL ) {return;}
+  if (upper_chain.right_most() == Vertex_handle() ) {return;}
 
   Neighbor_list* upper_list= upper_chain.up_list();
   Neighbor_list* lower_list= upper_chain.down_list();
@@ -597,7 +597,7 @@ set_infinite_faces()
     //both test are necessary because it may remain some  flat faces
     //in the upper chain.
     _tr->set_dimension(1);
-    newf = _tr->create_face(infinite, first->vertex(1), NULL);
+    newf = _tr->create_face(infinite, first->vertex(1), Face_handle());
     first = last = newf;
     infinite->set_face(first);
     typename Neighbor_list::iterator it = lower_list->begin();
@@ -606,13 +606,13 @@ set_infinite_faces()
       //turn the vertex [vww] into [wvNULL]
       fn->set_vertex(1, fn->vertex(0));
       fn->set_vertex(0, fn->vertex(2));
-      fn->set_vertex(2, Vertex_handle(NULL));
+      fn->set_vertex(2, Vertex_handle());
       fn->vertex(0)->set_face(fn);
       fn->set_neighbor(1,last);
       last->set_neighbor(0,fn);
       last = fn;
     }
-    fn = _tr->create_face(last->vertex(1), infinite,NULL);
+    fn = _tr->create_face(last->vertex(1), infinite, Vertex_handle());
     fn->vertex(0)->set_face(fn);
     fn->set_neighbor(1,last);
     last->set_neighbor(0,fn);
diff --git a/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_sweep_demo_2.h b/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_sweep_demo_2.h
index d4a49c8c40f..a99df1f8315 100644
--- a/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_sweep_demo_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Constrained_triangulation_sweep_demo_2.h
@@ -78,9 +78,9 @@ draw_new_faces(Vertex_handle v, Window_stream& W)
    drawing_mode dm=W.set_mode(leda_src_mode);
    W << BLUE;
    Face_handle f = v->face();
-   if (f == NULL ) { return;} //first point, no face
+   if (f == Face_handle() ) { return;} //first point, no face
    draw_face(f,W);
-   while( f->neighbor(1) != NULL ) {
+   while( f->neighbor(1) != Face_handle() ) {
      f = f->neighbor(1);
      draw_face(f,W);
    }
diff --git a/Packages/Triangulation_2/include/CGAL/Delaunay_triangulation_2.h b/Packages/Triangulation_2/include/CGAL/Delaunay_triangulation_2.h
index 13d92a945e6..41d158832b0 100644
--- a/Packages/Triangulation_2/include/CGAL/Delaunay_triangulation_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Delaunay_triangulation_2.h
@@ -69,13 +69,13 @@ public:
   bool is_valid(bool verbose = false, int level = 0) const;
 
   Vertex_handle
-  nearest_vertex(const Point& p, Face_handle f= Face_handle(NULL)) const;
+  nearest_vertex(const Point& p, Face_handle f= Face_handle()) const;
   
   bool does_conflict(const Point  &p, Face_handle fh) const;// deprecated
   bool test_conflict(const Point  &p, Face_handle fh) const;
   bool find_conflicts(const Point  &p,                //deprecated
 		      std::list<Face_handle>& conflicts,
-		      Face_handle start= Face_handle(NULL) ) const;
+		      Face_handle start= Face_handle() ) const;
   //  //template member functions, declared and defined at the end 
   // template <class OutputItFaces, class OutputItBoundaryEdges> 
   // std::pair<OutputItFaces,OutputItBoundaryEdges>
@@ -103,7 +103,7 @@ public:
   
   //INSERTION-REMOVAL
   Vertex_handle insert(const Point  &p, 
-		       Face_handle start = Face_handle(NULL) );
+		       Face_handle start = Face_handle() );
   Vertex_handle insert(const Point& p,
 		       Locate_type lt,
 		       Face_handle loc, int li );
@@ -160,7 +160,7 @@ public:
   get_conflicts_and_boundary(const Point  &p, 
 			     OutputItFaces fit, 
 			     OutputItBoundaryEdges eit,
-			     Face_handle start = Face_handle(NULL)) const {
+			     Face_handle start = Face_handle()) const {
     CGAL_triangulation_precondition( dimension() == 2);
     int li;
     Locate_type lt;
@@ -188,7 +188,7 @@ public:
   OutputItFaces
   get_conflicts (const Point  &p, 
 		 OutputItFaces fit, 
-		 Face_handle start= Face_handle(NULL)) const {
+		 Face_handle start= Face_handle()) const {
     std::pair<OutputItFaces,Emptyset_iterator> pp = 
       get_conflicts_and_boundary(p,fit,Emptyset_iterator(),start);
     return pp.first;
@@ -198,7 +198,7 @@ public:
   OutputItBoundaryEdges
   get_boundary_of_conflicts(const Point  &p, 
 			    OutputItBoundaryEdges eit, 
-			    Face_handle start= Face_handle(NULL)) const {
+			    Face_handle start= Face_handle()) const {
     std::pair<Emptyset_iterator, OutputItBoundaryEdges> pp = 
       get_conflicts_and_boundary(p,Emptyset_iterator(),eit,start);
     return pp.second;
@@ -281,7 +281,7 @@ nearest_vertex(const Point  &p, Face_handle f) const
 {
   switch (dimension()) {
   case 0:
-    if (number_of_vertices() == 0) return NULL;
+    if (number_of_vertices() == 0) return Vertex_handle();
     if (number_of_vertices() == 1) return finite_vertex();
     //break;
   case 1:
@@ -291,7 +291,7 @@ nearest_vertex(const Point  &p, Face_handle f) const
     return nearest_vertex_2D(p,f);
     //break;
   }
-  return NULL;
+  return Vertex_handle();
 }
   
 template < class Gt, class Tds >
@@ -300,7 +300,7 @@ Delaunay_triangulation_2<Gt,Tds>::
 nearest_vertex_2D(const Point& p, Face_handle f) const
 {
   CGAL_triangulation_precondition(dimension() == 2);
-  if (f == Face_handle(NULL)) f = locate(p);
+  if (f == Face_handle()) f = locate(p);
   else
     CGAL_triangulation_precondition(oriented_side(f,p)!=ON_NEGATIVE_SIDE);
 
@@ -488,7 +488,7 @@ void
 Delaunay_triangulation_2<Gt,Tds>::
 remove(Vertex_handle v )
 {
-  CGAL_triangulation_precondition( v != NULL);
+  CGAL_triangulation_precondition( v != Vertex_handle());
   CGAL_triangulation_precondition( !is_infinite(v));
         
   if ( dimension() <= 1) Triangulation::remove(v);
diff --git a/Packages/Triangulation_2/include/CGAL/Regular_triangulation_2.h b/Packages/Triangulation_2/include/CGAL/Regular_triangulation_2.h
index 3cf81a167c4..05e07f859a5 100644
--- a/Packages/Triangulation_2/include/CGAL/Regular_triangulation_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Regular_triangulation_2.h
@@ -192,7 +192,7 @@ public:
   // 						  OutputItBoundaryEdges eit,
   // 						  OutputItHiddenVertices vit,  
   // 						  Face_handle start = 
-  //                                                 Face_handle(NULL)) const;
+  //                                                 Face_handle()) const;
   // template <class OutputItFaces, class OutputItBoundaryEdges> 
   // std::pair<OutputItFaces,OutputItBoundaryEdges>
   // get_conflicts_and_boundary(const Point  &p, 
@@ -215,13 +215,13 @@ public:
   // 						OutputItBoundaryEdges eit, 
   // 						OutputItHiddenVertices vit,
   // 						Face_handle start=
-  //                                                Face_handle(NULL)) const;
+  //                                                Face_handle()) const;
   //   template <class OutputItHiddenVertices> 
   //   OutputItHiddenVertices
   //   get_hidden_vertices(const Weighted_point  &p, 
   // 						OutputItHiddenVertices vit,
   // 						Face_handle start= 
-  //                                                  Face_handle(NULL)) const;
+  //                                                  Face_handle()) const;
   
   // DUAL
   Bare_point dual (Face_handle f) const;
@@ -236,7 +236,7 @@ public:
   // Insertion, Deletion and Flip
   Vertex_handle push_back(const Weighted_point &p);
   Vertex_handle insert(const Weighted_point &p, 
-		       Face_handle f = Face_handle(NULL) );
+		       Face_handle f = Face_handle() );
   Vertex_handle insert(const Weighted_point &p,
 	 	       Locate_type  lt,
 		       Face_handle loc, int li );
@@ -244,7 +244,7 @@ public:
   Vertex_handle insert_in_edge(const Weighted_point &p, Face_handle f, int i);
   void flip(Face_handle f, int i);
   void remove_degree_3(Vertex_handle v, 
-		       Face_handle f = Face_handle(NULL));
+		       Face_handle f = Face_handle());
   void remove(Vertex_handle v);
 
   All_vertices_iterator all_vertices_begin () const;
@@ -332,7 +332,7 @@ public:
 						  OutputItBoundaryEdges eit,
 						  OutputItHiddenVertices vit,
 						  Face_handle start = 
-						  Face_handle(NULL)) const
+						  Face_handle()) const
     {
       CGAL_triangulation_precondition( dimension() == 2);
       int li;
@@ -420,7 +420,7 @@ public:
   get_conflicts_and_boundary (const Weighted_point  &p, 
 			      OutputItFaces fit, 
 			      OutputItBoundaryEdges eit,
-			      Face_handle start = Face_handle(NULL)) const
+			      Face_handle start = Face_handle()) const
     {
       Triple<OutputItFaces,OutputItBoundaryEdges,Emptyset_iterator>
 	pp = 
@@ -435,7 +435,7 @@ public:
 				    OutputItFaces fit, 
 				    OutputItHiddenVertices vit,
 				    Face_handle start = 
-				    Face_handle(NULL)) const
+				    Face_handle()) const
     {
       Triple<OutputItFaces, Emptyset_iterator,OutputItHiddenVertices> 
 	pp = 
@@ -453,7 +453,7 @@ public:
 						OutputItBoundaryEdges eit, 
 						OutputItHiddenVertices vit,
 						Face_handle start = 
-						Face_handle(NULL)) const
+						Face_handle()) const
     {
       Triple<Emptyset_iterator,OutputItBoundaryEdges,
 	OutputItHiddenVertices> 
@@ -469,7 +469,7 @@ public:
   OutputItFaces
   get_conflicts (const Weighted_point  &p, 
 		 OutputItFaces fit, 
-		 Face_handle start= Face_handle(NULL)) const
+		 Face_handle start= Face_handle()) const
     {
       Triple<OutputItFaces,Emptyset_iterator,Emptyset_iterator>
 	pp = 
@@ -484,7 +484,7 @@ public:
   OutputItBoundaryEdges
   get_boundary_of_conflicts(const Weighted_point  &p, 
 			    OutputItBoundaryEdges eit, 
-			    Face_handle start= Face_handle(NULL)) const
+			    Face_handle start= Face_handle()) const
     {    
       Triple<Emptyset_iterator, OutputItBoundaryEdges,Emptyset_iterator>
 	pp = 
@@ -498,7 +498,7 @@ public:
   template <class OutputItHiddenVertices> 
   OutputItHiddenVertices 
   get_hidden_vertices(const Weighted_point  &p, OutputItHiddenVertices vit,
-		      Face_handle start= Face_handle(NULL)) const
+		      Face_handle start= Face_handle()) const
     {
       Triple<Emptyset_iterator,Emptyset_iterator,
 	OutputItHiddenVertices> 
@@ -999,7 +999,7 @@ Regular_triangulation_2<Gt,Tds>::
 insert(const Weighted_point &p, Locate_type lt, Face_handle loc, int li) 
 {
   if (number_of_vertices() <= 1) return Base::insert(p);
-  Vertex_handle v(NULL);
+  Vertex_handle v();
   Oriented_side os;
   switch (lt) {
   case Base::VERTEX:
@@ -1193,7 +1193,7 @@ void
 Regular_triangulation_2<Gt,Tds>::
 remove_degree_3(Vertex_handle v, Face_handle f) 
 {
-  if (f == Face_handle(NULL))    f=v->face();
+  if (f == Face_handle())    f=v->face();
   update_hidden_points_3_1(f, f->neighbor( cw(f->index(v))),
 			   f->neighbor(ccw(f->index(v))));
   Triangulation_2<Gt,Tds>::remove_degree_3(v,f);
@@ -1222,7 +1222,7 @@ void
 Regular_triangulation_2<Gt,Tds>::
 remove(Vertex_handle v )
 {
-  CGAL_triangulation_precondition( v != NULL );
+  CGAL_triangulation_precondition( v != Vertex_handle() );
   CGAL_triangulation_precondition(!is_infinite(v));
    
   if (v->is_hidden()) {
diff --git a/Packages/Triangulation_2/include/CGAL/Triangulation_2.h b/Packages/Triangulation_2/include/CGAL/Triangulation_2.h
index cc1425ede88..d13473181e4 100644
--- a/Packages/Triangulation_2/include/CGAL/Triangulation_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Triangulation_2.h
@@ -251,7 +251,7 @@ public:
   Vertex_handle insert_in_face(const Point& p, Face_handle f);
   Vertex_handle insert_outside_convex_hull(const Point& p, Face_handle f);
   Vertex_handle insert_outside_affine_hull(const Point& p);
-  Vertex_handle insert(const Point &p, Face_handle start = Face_handle(NULL) );
+  Vertex_handle insert(const Point &p, Face_handle start = Face_handle() );
   Vertex_handle insert(const Point& p,
 		       Locate_type lt,
 		       Face_handle loc, int li );
@@ -259,7 +259,7 @@ public:
 //   int insert(InputIterator first, InputIterator last);
   Vertex_handle push_back(const Point& a);
  
-  void remove_degree_3(Vertex_handle  v, Face_handle f = Face_handle(NULL));
+  void remove_degree_3(Vertex_handle  v, Face_handle f = Face_handle());
   void remove_first(Vertex_handle  v);
   void remove_second(Vertex_handle v);
   void remove(Vertex_handle  v);
@@ -276,11 +276,11 @@ public:
   locate(const Point& p,
 	 Locate_type& lt,
 	 int& li,
-	 Face_handle start = Face_handle(NULL)) const;
+	 Face_handle start = Face_handle()) const;
 
   Face_handle
   locate(const Point &p,
-	 Face_handle start = Face_handle(NULL)) const;
+	 Face_handle start = Face_handle()) const;
     
 
   
@@ -310,15 +310,15 @@ public:
   Vertex_iterator vertices_end() const {return finite_vertices_end();}
 
   Face_circulator incident_faces( Vertex_handle v, 
-				  Face_handle f = Face_handle(NULL)) const;
+				  Face_handle f = Face_handle()) const;
   Vertex_circulator incident_vertices(Vertex_handle v,
-				      Face_handle f = Face_handle(NULL)) const;
+				      Face_handle f = Face_handle()) const;
   Edge_circulator incident_edges(Vertex_handle v,
-				 Face_handle f = Face_handle(NULL)) const;
+				 Face_handle f = Face_handle()) const;
  
   Line_face_circulator    line_walk(const Point& p,
 				    const Point& q,
-				    Face_handle f = Face_handle(NULL)) const;
+				    Face_handle f = Face_handle()) const;
 
  // TO DEBUG
   void show_all() const;
@@ -829,7 +829,7 @@ void
 Triangulation_2<Gt, Tds>::
 flip(Face_handle f, int i)
 {
-  CGAL_triangulation_precondition ( f != NULL );
+  CGAL_triangulation_precondition ( f != Face_handle() );
   CGAL_triangulation_precondition (i == 0 || i == 1 || i == 2);
   CGAL_triangulation_precondition( dimension()==2); 
     
@@ -1050,7 +1050,7 @@ insert(const Point& p, Locate_type lt, Face_handle loc, int li)
     return loc->vertex(li);
   }
   CGAL_triangulation_assertion(false);  // locate step failed
-  return Vertex_handle(NULL);
+  return Vertex_handle();
 }
 
 
@@ -1068,7 +1068,7 @@ inline void
 Triangulation_2<Gt,Tds>::
 remove_degree_3(Vertex_handle  v, Face_handle f)
 {
-  if (f == Face_handle(NULL)) f=v->face();
+  if (f == Face_handle()) f=v->face();
   _tds.remove_degree_3(v, f);
   return;
 }
@@ -1096,7 +1096,7 @@ void
 Triangulation_2<Gt,Tds>::      
 remove(Vertex_handle  v)
 {
-  CGAL_triangulation_precondition( v != NULL);
+  CGAL_triangulation_precondition( v != Verex_handle());
   CGAL_triangulation_precondition( !is_infinite(v));
     
   if  (number_of_vertices() == 1)     remove_first(v);
@@ -1184,7 +1184,7 @@ make_hole ( Vertex_handle v, std::list<Edge> & hole)
     if( vv->face()==  f) vv->set_face(fn);
     vv = f->vertex(ccw(i));
     if( vv->face()== f) vv->set_face(fn);
-    fn->set_neighbor(in, NULL);
+    fn->set_neighbor(in, Face_handle());
     hole.push_back(Edge(fn,in));
     to_delete.push_back(f);
   }
@@ -1566,7 +1566,7 @@ march_locate_1D(const Point& t,
   if(pqt == RIGHT_TURN || pqt == LEFT_TURN) {
     lt = OUTSIDE_AFFINE_HULL;
     li = 4 ;// should not be used
-    return Face_handle(NULL);
+    return Face_handle();
   }
 
   int i= f->index(ff);
@@ -1614,7 +1614,7 @@ march_locate_1D(const Point& t,
     }
   }
   CGAL_triangulation_assertion(false);
-  return Face_handle(NULL);
+  return Face_handle();
 }
 
 template <class Gt, class Tds >    
@@ -1703,13 +1703,13 @@ locate(const Point& p,
       }
       li = 4; // li should not be used in this case
     }
-    return NULL;
+    return Face_handle();
   }
   if(dimension() == 1){
     return march_locate_1D(p, lt, li);
   }
     
-  if(start == NULL){
+  if(start == Face_handle()){
     start = infinite_face()->
       neighbor(infinite_face()->index(infinite_vertex()));
   }else if(is_infinite(start)){
@@ -1891,7 +1891,7 @@ line_walk(const Point& p, const Point& q,  Face_handle f) const
 {
   CGAL_triangulation_precondition( (dimension() == 2) && 
 				! xy_equal(p,q));
-  Line_face_circulator lfc = (f == NULL)
+  Line_face_circulator lfc = (f == Face_handle())
     ? Line_face_circulator(p, q, this)
     : Line_face_circulator(p, q, f, this);
     
diff --git a/Packages/Triangulation_2/include/CGAL/Triangulation_ds_circulators_2.h b/Packages/Triangulation_2/include/CGAL/Triangulation_ds_circulators_2.h
index 2de021c9185..96b37c767db 100644
--- a/Packages/Triangulation_2/include/CGAL/Triangulation_ds_circulators_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Triangulation_ds_circulators_2.h
@@ -86,14 +86,16 @@ public:
   Face&
   operator*() const
   {
-    CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+    CGAL_triangulation_precondition(pos != Face_handle() && 
+				 _v != Vertex_handle());
     return *pos;
   }
 
   Face*
   operator->() const
   {
-    CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+    CGAL_triangulation_precondition(pos != Face_handle() && 
+				 _v != Vertex_handle());
     return &*pos;
   }
   
@@ -164,14 +166,16 @@ public:
   Vertex&
   operator*() const
   {
-    CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+    CGAL_triangulation_precondition(pos != Face_handle() && 
+				 _v != Vertex_handle());
     return *(pos->vertex(_ri));
   }
 
   Vertex*
   operator->() const
   {
-    CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+    CGAL_triangulation_precondition(pos != Face_handle() && 
+				 _v != Vertex_handle());
     return &*(pos->vertex(_ri));
   }
 
@@ -222,11 +226,11 @@ private:
 
 public:
   Triangulation_ds_edge_circulator_2()
-    : _v(NULL), pos(NULL)
+    : _v(), pos()
   {}
             
   Triangulation_ds_edge_circulator_2( Vertex_handle v, 
-				      Face_handle f = NULL);
+				      Face_handle f = Face_handle());
 
   Edge_circulator& operator++();
   Edge_circulator operator++(int);
@@ -259,11 +263,11 @@ Triangulation_ds_face_circulator_2<Tds> ::
 Triangulation_ds_face_circulator_2(Vertex_handle v, Face_handle f)
   : _v(v), pos(f)
 {
-  if (_v == NULL) pos = NULL;
-  else if ( pos == NULL) pos = v->face();
+  if (_v == Vertex_handle()) pos = Face_handle();
+  else if ( pos == Face_handle()) pos = v->face();
 
-  if (pos == NULL || pos->dimension() < 2) { 
-    _v = NULL ; pos = NULL; return;}
+  if (pos ==  Face_handle()|| pos->dimension() < 2) { 
+    _v =  Vertex_handle() ; pos = Face_handle(); return;}
   else CGAL_triangulation_precondition( pos->has_vertex(v));
 }
  
@@ -284,7 +288,8 @@ Triangulation_ds_face_circulator_2<Tds>&
 Triangulation_ds_face_circulator_2<Tds> ::
 operator++()
 {
-  CGAL_triangulation_precondition( (pos != NULL) && (_v != NULL) );
+  CGAL_triangulation_precondition( (pos != Face_handle() && 
+				 _v != Vertex_handle());
   int i = pos->index(_v);
   pos = pos->neighbor(ccw(i));
   return *this;
@@ -295,7 +300,8 @@ Triangulation_ds_face_circulator_2<Tds>
 Triangulation_ds_face_circulator_2<Tds> ::
 operator++(int)
 {
-  CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+  CGAL_triangulation_precondition(pos != Face_handle() && 
+				 _v != Vertex_handle());
   Face_circulator tmp(*this);
   ++(*this);
   return tmp;
@@ -306,7 +312,8 @@ Triangulation_ds_face_circulator_2<Tds>&
 Triangulation_ds_face_circulator_2<Tds> ::
 operator--()
 {
-   CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+   CGAL_triangulation_precondition(pos != Face_handle() && 
+				 _v != Vertex_handle());
    int i = pos->index(_v);
    pos = pos->neighbor(cw(i));
    return *this;
@@ -317,7 +324,8 @@ Triangulation_ds_face_circulator_2<Tds>
 Triangulation_ds_face_circulator_2<Tds> ::
 operator--(int)
 {
-  CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+  CGAL_triangulation_precondition(pos != Face_handle() && 
+				 _v != Vertex_handle());
   Face_circulator tmp(*this);
   --(*this);
   return tmp;
@@ -344,7 +352,7 @@ inline bool
 Triangulation_ds_face_circulator_2<Tds> ::
 is_empty() const
 {    
-return ((_v == NULL) || (pos == NULL));  
+return (_v == Vertex_handle() ||  pos == Face_handle() );  
 }
 
 template < class Tds >
@@ -353,7 +361,7 @@ Triangulation_ds_face_circulator_2<Tds> ::
 operator==(CGAL_NULL_TYPE CGAL_triangulation_assertion_code(n)) const
 {
   CGAL_triangulation_assertion( n == NULL);
-  return (_v == NULL || pos == NULL);
+  return (_v == Vertex_handle() ||  pos == Face_handle() );  
 }
         
 template < class Tds >
@@ -419,7 +427,7 @@ Triangulation_ds_vertex_circulator_2<Tds>&
 Triangulation_ds_vertex_circulator_2<Tds> ::
 operator--()
 {
-  CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+  CGAL_triangulation_precondition(pos != Face_handle() && _v != Vertex_handle());
   int i = pos->index(_v);
     
   if (pos->dimension() == 1) { 
@@ -492,11 +500,11 @@ Triangulation_ds_edge_circulator_2<Tds> ::
 Triangulation_ds_edge_circulator_2(Vertex_handle v, Face_handle f)
     : _v(v), pos(f)
 {
-  if (_v == NULL) { pos = NULL;}
-  else if (pos==NULL) {pos = v->face();}
+  if (_v == Vertex_handle()) { pos = Face_handle();}
+  else if (pos==Face_handle()) {pos = v->face();}
  
-  if (pos == NULL || pos->dimension() < 1){
-    _v = NULL; pos = NULL;return;}
+  if (pos == Face_handle() || pos->dimension() < 1){
+    _v = Vertex_handle(); pos = Face_handle();return;}
   int i = pos->index(_v);
   if (pos->dimension() == 2) {_ri = ccw(i);}
   else {_ri = 2;}
@@ -508,7 +516,7 @@ Triangulation_ds_edge_circulator_2<Tds>&
 Triangulation_ds_edge_circulator_2<Tds> ::
 operator++()
 {
-  CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+  CGAL_triangulation_precondition(pos != Face_handle() && _v != Vertex_handle());
   int i = pos->index(_v);
   if (pos->dimension() == 1) { 
     pos = pos->neighbor(1-i);
@@ -537,7 +545,7 @@ Triangulation_ds_edge_circulator_2<Tds>&
 Triangulation_ds_edge_circulator_2<Tds> ::
 operator--()
 {
-  CGAL_triangulation_precondition(pos != NULL && _v != NULL);
+  CGAL_triangulation_precondition(pos != Face_handle() && _v != Vertex_handle());
   int i = pos->index(_v);
 
   if (pos->dimension() == 1) { 
@@ -583,7 +591,7 @@ inline bool
 Triangulation_ds_edge_circulator_2<Tds> ::
 is_empty() const
 {
-  return (_v == NULL || pos == NULL);
+  return (_v == Vertex_handle() || pos == Face_handle());
 }
 
 template < class Tds >
@@ -592,7 +600,7 @@ Triangulation_ds_edge_circulator_2<Tds> ::
 operator==(CGAL_NULL_TYPE CGAL_triangulation_assertion_code(n)) const
 {
   CGAL_triangulation_assertion( n == NULL);
-  return (_v == NULL || pos == NULL);
+  return (v == Vertex_handle() || pos == Face_handle());
 }
                
 template < class Tds >
diff --git a/Packages/Triangulation_2/include/CGAL/Triangulation_ds_face_2.h b/Packages/Triangulation_2/include/CGAL/Triangulation_ds_face_2.h
index 05a0096ac5b..f4b59720d8c 100644
--- a/Packages/Triangulation_2/include/CGAL/Triangulation_ds_face_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Triangulation_ds_face_2.h
@@ -72,7 +72,7 @@ typename Triangulation_ds_face_2<Fb>::Vertex_handle
 Triangulation_ds_face_2<Fb>::
 mirror_vertex(int i) const
 {
-  CGAL_triangulation_precondition ( neighbor(i) != NULL &&  dimension() >= 1);
+  CGAL_triangulation_precondition ( neighbor(i) != Face_handle() &&  dimension() >= 1);
   //return neighbor(i)->vertex(neighbor(i)->index(this->handle()));
   return neighbor(i)->vertex(mirror_index(i));
 }
diff --git a/Packages/Triangulation_2/include/CGAL/Triangulation_ds_vertex_2.h b/Packages/Triangulation_2/include/CGAL/Triangulation_ds_vertex_2.h
index 72dfc1af10d..066e1315a35 100644
--- a/Packages/Triangulation_2/include/CGAL/Triangulation_ds_vertex_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Triangulation_ds_vertex_2.h
@@ -110,7 +110,7 @@ is_valid(bool verbose, int level)
 {
   bool result = Vb::is_valid(verbose, level);
   CGAL_triangulation_assertion(result);
-  if (face() != NULL) { // face==NULL if dim <0
+  if (face() != Face_handle()) { // face==NULL if dim <0
     result = result && ( &*face()->vertex(0) == this ||
 			 &*face()->vertex(1) == this ||
 			 &*face()->vertex(2) == this );
diff --git a/Packages/Triangulation_2/include/CGAL/Triangulation_hierarchy_2.h b/Packages/Triangulation_2/include/CGAL/Triangulation_hierarchy_2.h
index ef071c2e826..62271c2d315 100644
--- a/Packages/Triangulation_2/include/CGAL/Triangulation_hierarchy_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Triangulation_hierarchy_2.h
@@ -73,7 +73,7 @@ public:
   bool is_valid(bool verbose = false, int level = 0) const;
 
   // INSERT REMOVE
-  Vertex_handle insert(const Point &p, Face_handle start = Face_handle(NULL) );
+  Vertex_handle insert(const Point &p, Face_handle start = Face_handle() );
   Vertex_handle insert(const Point& p,
 		       Locate_type lt,
 		       Face_handle loc, int li );
@@ -100,11 +100,11 @@ public:
   locate(const Point& p,
 	 Locate_type& lt,
 	 int& li,
-	 Face_handle start = Face_handle(NULL)) const;
+	 Face_handle start = Face_handle()) const;
 
   Face_handle
   locate(const Point &p,
-	 Face_handle start = Face_handle(NULL)) const;
+	 Face_handle start = Face_handle()) const;
 
 private:
   void  locate_in_all(const Point& p,
@@ -168,7 +168,7 @@ copy_triangulation(const Triangulation_hierarchy_2<Tr> &tr)
   {
     for( Finite_vertices_iterator it=hierarchy[0]->finite_vertices_begin(); 
 	 it != hierarchy[0]->finite_vertices_end(); ++it) {
-      if (it->up() != NULL) V[ it->up()->down() ] = it;
+      if (it->up() != Vertex_handle()) V[ it->up()->down() ] = it;
     }
   }
   {
@@ -180,7 +180,7 @@ copy_triangulation(const Triangulation_hierarchy_2<Tr> &tr)
 	// make reverse link
 	it->down()->set_up(it);
 	// make map for next level
-	if (it->up()!= NULL ) V[ it->up()->down() ] = it;
+	if (it->up()!=  Vertex_handle() ) V[ it->up()->down() ] = it;
       }
     }
   }
@@ -238,7 +238,7 @@ is_valid(bool verbose, int level) const
     //verify that lower level has no down pointers
   for( it = hierarchy[0]->finite_vertices_begin(); 
        it != hierarchy[0]->finite_vertices_end(); ++it) 
-    result = result && ( it->down() == NULL );
+    result = result && ( it->down() ==   Vertex_handle());
   //verify that other levels have down pointer and reciprocal link is fine
   for(i=1;i<Triangulation_hierarchy_2__maxlevel;++i)
     for( it = hierarchy[i]->finite_vertices_begin(); 
@@ -249,7 +249,7 @@ is_valid(bool verbose, int level) const
   for(i=0;i<Triangulation_hierarchy_2__maxlevel-1;++i)
     for( it = hierarchy[i]->finite_vertices_begin(); 
 	 it != hierarchy[i]->finite_vertices_end(); ++it) 
-      result = result && ( it->up() == NULL ||
+      result = result && ( it->up() ==  Vertex_handle() ||
 	        &*it == &*(it->up())->down() );
   return result;
 }
@@ -334,7 +334,7 @@ remove(Vertex_handle v )
   int l = 0 ;
   while(1){
     hierarchy[l++]->remove(v);
-    if (u == NULL) break; 
+    if (u ==  Vertex_handle()) break; 
     if(l>Triangulation_hierarchy_2__maxlevel) break;
     v=u; u=v->up();
   }
@@ -393,8 +393,8 @@ locate_in_all(const Point& p,
     int& li,
     Face_handle pos[Triangulation_hierarchy_2__maxlevel]) const
 {
-  Face_handle position(NULL);
-  Vertex_handle nearest(NULL);
+  Face_handle position();
+  Vertex_handle nearest();
   int level  = Triangulation_hierarchy_2__maxlevel;
   typename Geom_traits::Compare_distance_2 
     closer = geom_traits().compare_distance_2_object();
diff --git a/Packages/Triangulation_2/include/CGAL/Triangulation_line_face_circulator_2.h b/Packages/Triangulation_2/include/CGAL/Triangulation_line_face_circulator_2.h
index 6c971f803c9..1b12bb9aaab 100644
--- a/Packages/Triangulation_2/include/CGAL/Triangulation_line_face_circulator_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Triangulation_line_face_circulator_2.h
@@ -69,7 +69,7 @@ private:
             
 public:
   Triangulation_line_face_circulator_2()
-    : pos(NULL), _tr(NULL), s(undefined), i(-1)
+    : pos(), _tr(NULL), s(undefined), i(-1)
     {}
             
   Triangulation_line_face_circulator_2(Vertex_handle v,
@@ -157,7 +157,7 @@ Triangulation_line_face_circulator_2<Gt,Tds>::
 Triangulation_line_face_circulator_2(Vertex_handle v,
 				     const Triangulation_2<Gt,Tds>* tr,
 				     const Point& dir)
-  :pos(NULL), _tr(tr), s(undefined)
+  :pos(), _tr(tr), s(undefined)
   // begin at the face incident to v, traversed by the ray from v to
   // dir 
   // or null iterator
@@ -239,7 +239,7 @@ Triangulation_line_face_circulator_2<Gt,Tds>::
 Triangulation_line_face_circulator_2(const Point& pp,
 				     const Point& qq,
 				     const Triangulation_2<Gt,Tds> * t)
-     : pos(NULL), _tr(t), s(undefined), p(pp), q(qq)
+     : pos(), _tr(t), s(undefined), p(pp), q(qq)
   //begins at the  first finite face traversed be the oriented line pq
 {
   Vertex_handle inf = _tr->infinite_vertex();
@@ -425,7 +425,7 @@ void
 Triangulation_line_face_circulator_2<Gt,Tds>::
 increment()
 {
-  CGAL_triangulation_precondition(pos != CGAL_CIRC_NULL);
+  CGAL_triangulation_precondition(pos != Face_handle());
   if(s == vertex_vertex || s == edge_vertex) {
     Orientation o;
     do{
@@ -478,7 +478,7 @@ void
 Triangulation_line_face_circulator_2<Gt,Tds>::             
 decrement()
 {
-  CGAL_triangulation_precondition(pos != CGAL_CIRC_NULL);
+  CGAL_triangulation_precondition(pos != Face_handle());
   if(s == vertex_vertex || s == vertex_edge) {
     if(s == vertex_vertex){
       i = cw(i);
@@ -598,7 +598,7 @@ Triangulation_line_face_circulator_2<Gt,Tds>&
 Triangulation_line_face_circulator_2<Gt,Tds>::
 operator++()
 {
-  CGAL_triangulation_precondition( pos != CGAL_CIRC_NULL) ;
+  CGAL_triangulation_precondition( pos != Face_handle()) ;
   increment();
   return *this;
 }
@@ -609,7 +609,7 @@ Triangulation_line_face_circulator_2<Gt,Tds>&
 Triangulation_line_face_circulator_2<Gt,Tds>::            
 operator--()
 {
-  CGAL_triangulation_precondition(*this != CGAL_CIRC_NULL) ;
+  CGAL_triangulation_precondition(pos != Face_handle()) ;
   decrement();
   return *this;
 }
@@ -641,8 +641,8 @@ inline bool
 Triangulation_line_face_circulator_2<Gt,Tds>::    
 operator==(const Line_face_circulator& lfc) const
 {
-  CGAL_triangulation_precondition( *this != CGAL_CIRC_NULL &&
-				   lfc != CGAL_CIRC_NULL);
+  CGAL_triangulation_precondition( pos != Face_handle() &&
+			       lfc.pos != Face_handle());
   return ( pos == lfc.pos &&  _tr == lfc._tr &&
             s== lfc.s && p==lfc.p && q==lfc.q);
 }
@@ -660,7 +660,7 @@ inline bool
 Triangulation_line_face_circulator_2<Gt,Tds>::   
 is_empty() const
 {
-  return pos == (CGAL_CIRC_NULL);
+  return pos == Face_handle();
 }
 
 template < class Gt, class Tds >
@@ -669,7 +669,7 @@ Triangulation_line_face_circulator_2<Gt,Tds>::
 operator==(CGAL_NULL_TYPE CGAL_triangulation_assertion_code(n)) const
 {
   CGAL_triangulation_assertion( n == NULL);
-  return pos ==(CGAL_CIRC_NULL);
+  return pos == Face_handle();
 }
             
 template < class Gt, class Tds >