  This text surveys classical and recent results in the field of amenability of
groups, from a combinatorial standpoint. It has served as the support of
courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure.
  The goals of the text are (1) to be as self-contained as possible, so as to
serve as a good introduction for newcomers to the field; (2) to stress the use
of combinatorial tools, in collaboration with functional analysis, probability
etc., with discrete groups in focus; (3) to consider from the beginning the
more general notion of amenable actions; (4) to describe recent classes of
examples, and in particular groups acting on Cantor sets and topological full
groups.
