The Gauss-Bonnet Theorem states that a certain integral of a two-dimensional Riemannian manifold's curvature is equal to the manifold's Euler characteristic.  I shall prove this theorem in three (or maybe more) ways, using three different definitions of curvature, and three different definitions of the Euler characteristic.  But are these "really" all the same definition?  And are these "really" all the same proof?