In this paper we study the problem of flipping edges in triangulations of polygons and point sets. We prove that if a polygon Qn has k reflex vertices, then any triangulation T of Qn can be transformed to another triangulation T' of Qn by flipping at most O(n + k2) edges. We produce examples of polygons with triangulations T and T' such that to transform T to T' requires O(n2) flips. These results are then extended to triangulations of point sets. We also show that any triangulation of an n point set contains at least (n - 4) / 2 edges that can be flipped.