We describe Baez and Dolan's cobordism hypothesis - a deep connection between topological quantum field theory, higher categories, and manifolds. Physically, this encodes the idea that quantum field theories, even "topological" ones, should be local: no matter how we cut up the spacetime on which they are defined in order to perform the path integral, the net result must be the same. Recently, this hypothesis was formulated and proved by Jacob Lurie using the tools of homotopy theory. We describe the version of the hypothesis he proved. Finally, we touch on Freed, Hopkins, Lurie and Teleman's recent work on Chern-Simons theory, and on Urs Schreiber's ideas for using Lurie's toolkit in full-fledged quantum field theory.