We show how to triangulate polygonal regions---adding extra vertices as necessary--- with triangles of guaranteed quality. Using only O(n) triangles, we can guarantee that the smallest height (shortest dimension) of a triangle in a triangulation of an n-vertex polygon (with holes) is a constant fraction of the largest possible. For simple polygons, using O(n log n) triangles, we can guarantee that the largest angle is no greater than 150 ffi . This bound increases to O(n 3=2 ) triangles for the case of polygons with holes. We can add the guarantee on smallest height to these no-large-angle results, without increasing the asymptotic complexity of the triangulation. Finally we give a nonobtuse triangulation algorithm for convex polygons that uses O(n 1:85 ) triangles. Keywords: Computational geometry, mesh generation, triangulation, angle condition. 1. Introduction There have been a number of recent papers on the general problem of triangulating a planar point set or pol..