The Gauss-Bonnet theorem is a beautiful result about the geometry and topology of closed oriented surfaces, and relates a geometric quantity, curvature, to a topological quantity, the Euler characteristic. The idea that curvature should carry topological information can be pushed quite far. In this talk I will talk about one way to construct characteristic classes of principal (or vector) bundles using curvature, and will also give a construction of Chern classes of a complex vector bundle in terms of curvature.