  Shortly after Szemer\'edi's proof that a set of positive upper density
contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof
of this theorem using ergodic theory. This gave rise to the field of ergodic
Ramsey Theory, in which problems motivated by additive combinatorics are proven
using ergodic theory. Ergodic Ramsey Theory has since produced combinatorial
results, some of which have yet to be obtained by other means, and has also
given a deeper understanding of the structure of measure preserving systems. We
outline the ergodic theory background needed to understand these results, with
an emphasis on recent developments in ergodic theory and the relation to recent
developments in additive combinatorics.
  These notes are based on four lectures given during the School on Additive
Combinatorics at the Centre de Recherches Math\'ematiques, Montreal in April,
2006. The talks were aimed at an audience without background in ergodic theory.
No attempt is made to include complete proofs of all statements and often the
reader is referred to the original sources. Many of the proofs included are
classic, included as an indication of which ingredients play a role in the
developments of the past ten years.
