  This text is an introduction to equivariant cohomology, a classical tool for
topological transformation groups, and to equivariant intersection theory, a
much more recent topic initiated by D. Edidin and W. Graham. It is based on
lectures given at Montr\'eal is Summer 1997.
  Our main aim is to obtain explicit descriptions of cohomology or Chow rings
of certain manifolds with group actions which arise in representation theory,
e.g. homogeneous spaces and their compactifications.
  As another appplication of equivariant intersection theory, we obtain simple
versions of criteria for smoothness or rational smoothness of Schubert
varieties (due to Kumar, Carrell-Peterson and Arabia) whose statements and
proofs become quite transparent in this framework.
