In this paper it is shown that there is some constant c, such that for any polygon, with or without holes, with w concave vertices, the length of any greedy triangulation of the polygon is not longer than c x (w + 1) times the length of a minimum weight triangulation of the polygon (under the assumption that no three vertices lie on the same line). A low approximation constant is proved for interesting classes of polygons. On the other hand, it is shown that for every integer n greater than 3, there exists some set S of n points in the plane, such that the greedy triangulation of S is W(n1/2) times longer than the minimum weight triangulation (this improves the previously known W(n1/3) lower bound). Finally, a simple linear-time algorithm is presented and analyzed for computing greedy triangulations of polygons with the so called semi-circle property.