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 C*-simple 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 viewed as an extreme case of non-amenability. The basic examples are described, and several open problems are formulated.