  In 1975 Szemer\'edi proved the long-standing conjecture of Erd\H{o}s and
Tur\'an that any subset of $\bbZ$ having positive upper Banach density contains
arbitrarily long arithmetic progressions. Szemer\'edi's proof was entirely
combinatorial, but two years later Furstenberg gave a quite different proof of
Szemer\'edi's Theorem by first showing its equivalence to an ergodic-theoretic
assertion of multiple recurrence, and then bringing new machinery in ergodic
theory to bear on proving that. His ergodic-theoretic approach subsequently
yielded several other results in extremal combinatorics, as well as revealing a
range of new phenomena according to which the structures of
probability-preserving systems can be described and classified.
  In this work I survey some recent advances in understanding these
ergodic-theoretic structures. It contains proofs of the norm convergence of the
`nonconventional' ergodic averages that underly Furstenberg's approach to
variants of Szemer\'edi's Theorem, and of two of the recurrence theorems of
Furstenberg and Katznelson: the Multidimensional Multiple Recurrence Theorem,
which implies a multidimensional generalization of Szemer\'edi's Theorem; and a
density version of the Hales-Jewett Theorem of Ramsey Theory.
