I will begin with an overview of results (joint with B. Host) on nonconventional ergodic averages, considering averages along arithmetic progressions, polynomials and combinatorial cubes. For each average, the L^2 convergence is determined by a certain factor of the system that is nilpotent. I will also discuss more recent results on the limit in certain systems and partial results on commuting transformations (joint with N. Frantzikinakis).