In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least \(\lceil (n-4)/2 \rceil\) edges that can be flipped. We also prove that O(n + k 2 ) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n 2 ) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips.