A 'topological quantum field theory' (TQFT) is a functor from the n-dimensional cobordism category (objects are (n-1)-dimensional manifolds, morphisms are cobordisms) to vector spaces. These were introduced by Atiyah and Segal in the 90's, when they were trying to make sense of what the physicists were doing. If one thinks hard about them, one finds that all sorts of higher category ideas naturally pop out. These ideas have deep connections to algebraic topology: for instance, the Freed-Teleman-Hopkins theorem about equivariant K-theory is a theorem about TQFT's. I will introduce the simplest non-trivial model, based on a finite group, and use it to give an introduction to (what is now called) 'extended' TQFT.