In 1977 Furstenberg gave a beautiful new proof of a famous
result of Szemeredi, asserting that any positive-density subset of the
integers contains arbitrarily long arithmetic progressions.  Unlike
Szemeredi's purely combinatorial proof, Furstenberg showed the
equivalence of this fact to an assertion of multiple recurrence in
ergodic theory.  In the years that followed, his work not only led to
several different generalizations of Szemeredi's Theorem -- some of
which have only very recently been proved by any other method – but
also prompted a search for a more detailed understanding of the
ergodic theoretic structures that govern multiple recurrence.  In this
talk I will sketch the connexion between ergodic theory and
Szemeredi's Theorem, and describe some of the structural results in
ergodic theory (with some simple examples) that are needed for the
various proofs of multiple recurrence (both Furstenberg's and other,
more recent proofs).