We will present a mathematical approach to classical and quantum field theories, in terms of smooth functors on geometric bordism categories. This allows the construc- tion of classifying spaces for such field theories, and a number of deep analogies with algebraic topology arises in the super symmetric case: Locality of a QFT leads to Mayer- Vietoris sequences, central charge corresponds to the degree (or twist) of a cohomology class, gauged field theories arise in equivariant cohomology and quantization corre- sponds to push-forward maps. For very small space-time dimensions, we will explain how these analogies become theorems by identifying classifying spaces of field theo- ries with those of de Rham cohomology respectively K-theory. We will report on joint work with Stephan Stolz and many others.