A smooth manifold is a topological structure that admits some standard calculus constructions. A vector bundle over a manifold is a smooth choice of a vector space at every point of the manifold. One motivation for such an object is that the differential of a map between two manifolds is a map between their tangent bundles. At first glance, vector bundles seem as though they should be Cartesian products of the manifold with a vector space. In general, this is false, and certain ``characteristic classes'' are the explicit obstructions. In this talk, we will construct these classes and discuss some theorems explaining why they are of interest to many mathematicians.