This course will be a gentle introduction to some of the technical tools that underlie recent approaches to Szemeredi's Theorem in additive combinatorics and associated questions about multiple averages in ergodic theory. We will begin with a synopsis of Furstenberg's work relating Szemeredi's Theorem to the ergodic theoretic phenomenon of multiple recurrence, and with some of the technical background needed for this connection to bear fruit. We will then quickly sketch Gowers' approach to Szemeredi's Theorem, which yielded the best-known bounds and pointed towards various extensions, such as the Green-Tao result for the primes. A key tool introduced by Gowers is a family of seminorms for functions on Z/n, together with an `inverse theorem' describing the structure of those functions for which these seminorms take large values. These seminorms have counterparts in ergodic theory, introduced by Host and Kra for their work on convergence of multiple averages, also along with an inverse theorem (closely related to another analysis by Ziegler, not in terms of seminorms). These ideas will be summarized with an emphasis on their commonalities. In both settings, the known inverse theorems give not only proofs of new or tightened results, but a much improved understanding of the structures responsible for them. Having done this, the course will go into more detail about the ergodic-theoretic inverse theorems. The classical case of triple averages will be treated carefully, and its generalization covered more tersely. Finally, we will sketch the difficulties of extending the above work, either additive combinatorial or ergodic theoretic, to results for higher-rank Abelian groups. Here the desired inverse theorems remain quite mysterious. Depending on time, we will finish with a sketch of some recent work on a simple `extremal' version of the inverse problem in higher dimensions, where one already finds considerable new structure.