This talk will be a general introduction to the theory of non-conventional ergodic averages. One example is the average of a product of functions evaluated along an arithmetic progression, such as the expressions appearing in Furstenberg's proof of Szemeredi's theorem. For each average studied, it suffices to prove the convergence for certain special systems, known as characteristic factors. To each of these systems, we associate a natural group of transformations that has the structure of a nilmanifold -- this is the "ergodic geometry" behind the convergence. If time allows, I will mention some combinatorial corollaries of the ergodic theorem.