Abstract: The study of the curvature of polyhedra goes back to Descartes. For a polyhedral surface M and a vertex v of Mv is the angle defect, which is 2\pi minus the sum of the angles at v in the polygons containing v. Descartes proved that the sum of the angle defects at the vertices of a convex polyhedron in R^3 is 4\pi, which is a polyhedral analog of the Gauss-Bonnet theorem. This talk, which will be entirely elementary, will discuss some of the geometric and combinatorial generalizations of the angle defect to higher dimensions and to non-manifolds.