Abstract: Solving a conjecture of Erdos and Turan, Szemeredi proved in the 1970's that a set of integers with positive upper density contains arithmetic progressions of arbitrary length. Furstenberg gave an ergodic theoretic proof of Szemeredi's Theorem based on multiple recurrence, leading to the study of "non-conventional" ergodic averages -- these are averaging theorems where even when the system is ergodic, the limit need not be constant. As an example, we prove the existence in L^2 of the limit of averages of the form 1/N \sum_n f(T^nx)g(T^{2n}x)h(T^{3n}x) as N tends to infinity. Closely related is averages along cubes, for example 1/N \sum_n f(T^nx)g(T^mx)h(T^{n+m}x). We can show that the analogous average (for example with 7 terms) exists in L^2. This gives a new combinatorial statement about subsets of integers of positive upper density. This is joint work with B. Host.