  The electronic ground state of a periodic system is usually described in
terms of extended Bloch orbitals, but an alternative representation in terms of
localized "Wannier functions" was introduced by Gregory Wannier in 1937. The
connection between the Bloch and Wannier representations is realized by
families of transformations in a continuous space of unitary matrices, carrying
a large degree of arbitrariness. Since 1997, methods have been developed that
allow one to iteratively transform the extended Bloch orbitals of a
first-principles calculation into a unique set of maximally localized Wannier
functions, accomplishing the solid-state equivalent of constructing localized
molecular orbitals, or "Boys orbitals" as previously known from the chemistry
literature. These developments are reviewed here, and a survey of the
applications of these methods is presented. This latter includes a description
of their use in analyzing the nature of chemical bonding, or as a local probe
of phenomena related to electric polarization and orbital magnetization.
Wannier interpolation schemes are also reviewed, by which quantities computed
on a coarse reciprocal-space mesh can be used to interpolate onto much finer
meshes at low cost, and applications in which Wannier functions are used as
efficient basis functions are discussed. Finally the construction and use of
Wannier functions outside the context of electronic-structure theory is
presented, for cases that include phonon excitations, photonic crystals, and
cold-atom optical lattices.
