  This is an expository book on unitary representations of topological groups,
and of several dual spaces, which are spaces of such representations up to some
equivalence. The most important notions are defined for topological groups, but
a special attention is paid to the case of discrete groups.
  The unitary dual of a group $G$ is the space of equivalence classes of its
irreducible unitary representations; it is both a topological space and a Borel
space. The primitive dual is the space of weak equivalence classes of unitary
irreducible representations. The normal quasi-dual is the space of
quasi-equivalence classes of traceable factor representations; it is
parametrized by characters, which can be finite or infinite.
  The theory is systematically illustrated by a series of specific examples:
Heisenberg groups, affine groups of infinite fields, solvable Baumslag-Solitar
groups, lamplighter groups, and general linear groups.
  Operator algebras play an important role in the exposition, in particular the
von Neumann algebras associated to a unitary representation and C*-algebras
associated to a locally compact group.
