  A countable group is C*-simple if its reduced C*-algebra is a simple algebra.
Since Powers recognised in 1975 that non-abelian free groups are C*-simple,
large classes of groups which appear naturally in geometry have been
identified, including non-elementary Gromov hyperbolic groups and lattices in
semisimple groups.
  In this exposition, C*-simplicity for countable groups is shown to be an
extreme case of non-amenability. The basic examples are described and several
open problems are formulated.
