  These notes are based on a course for a general audience given at the Centro
de Modeliamento Matem\'atico of the University of Chile, in December 2004.
  We study the mean convergence of multiple ergodic averages, that is, averages
of a product of functions taken at different times. We also describe the
relations between this area of ergodic theory and some classical and some
recent results in additive number theory.
