% \section{INTRODUCTION}
% \label{sec:intro}

% ---------------------------------------------------

In the independent-particle approximation, the electronic ground state
of a system is determined by specifying a set of one-particle
orbitals and their occupations.  For the case of periodic system, these one-particle
orbitals are normally taken to be the Bloch functions
$\psi_{n\kk}(\rr)$ that are labeled, according to Bloch's theorem, by
a crystal momentum $\bf k$ lying inside the Brillouin zone and a band
index $n$.  Although this choice is by far the most widely used in
electronic-structure calculations, alternative representations are
possible.  In particular, to arrive at the Wannier representation
\cite{wannier-pr37,kohn-pr59,cloizeaux-pr63}, one carries out a
unitary transformation from the Bloch functions to a set of localized
``Wannier functions'' (WFs) labeled by a cell index $\bf R$ and a band-like
%
index $n$, such that in a crystal the WFs at different $\bf R$ are
translational images of one another.  Unlike Bloch functions, 
WFs are not eigenstates of the Hamiltonian; in selecting
them, one trades off localization in energy for localization in space.

In the earlier solid-state theory literature, WFs
were typically introduced in order to carry out some formal
derivation -- for example, of the effective-mass treatment
of electron dynamics, or of an effective spin Hamiltonian -- 
but actual calculations of the WFs were rarely performed.
The history is rather different in the chemistry literature, where
``localized molecular orbitals'' (LMO's)
\cite{boys-rmp60,foster-rmp60a,foster-rmp60b,edmiston-rmp63,boys-66}
have played a significant role in computational chemistry since
its early days.  Chemists have emphasized that such a representation
can provide an insightful picture of the nature of
the chemical bond in a material --- otherwise missing from the
picture of extended eigenstates --- or can serve as a compact
basis set for high-accuracy calculations.

The actual implementation of Wannier's vision in the
context of first-principles electronic-structure calculations,
such as those carried out in the Kohn-Sham framework of
density-functional theory \cite{kohn-pr65},
has instead been slower to unfold.
A major reason for this is that WFs are
strongly non-unique. This is a consequence of
the phase indeterminacy that Bloch orbitals
$\psi_{n{\bf k}}$ have at every wavevector {\bf k} -- or, more
generally, the ``gauge'' indeterminacy associated with the freedom
to apply any arbitrary unitary transformation to
the occupied Bloch states at each $\kk$.  This second
indeterminacy is all the more troublesome in the common case of
degeneracy for the occupied bands at certain high-symmetry
points in the Brillouin zone, making a partition into
separate ``bands'', that could separately be transformed in
Wannier functions, problematic.  Therefore, even 
before one could attempt to
compute the WFs for a given material, one had first
to resolve the question of which states to use to compute
WFs.

An important development in this regard was the introduction
by \textcite{marzari-prb97}
of a ``maximal localization'' criterion for identyfing a unique
set of WFs for a given crystalline insulator.
The approach is similar in spirit to the construction of LMO's
in chemistry, but its implementation in the
solid-state context required significant developments, due to the
ill-conditioned nature of the position operator in periodic systems
\cite{nenciu-rmp91}, 
that was clarified in the context of the ``modern theory'' of polarization
\cite{king-smith-prb93,resta-rmp94}.
Marzari and Vanderbilt showed that the minimization of a 
localization functional corresponding to the sum of the
second-moment spread of each Wannier charge density about its own
center of charge was both formally attractive and computationally
tractable.  In a related
development, \textcite{souza-prb01} generalized
the method to handle the case in which one wants to construct a
set of WFs that spans a subspace containing, e.g.,
the partially occupied bands of a metal.

These developments touched off a transformational shift in which
the computational electronic-structure community started constructing
maximally-localized WFs (MLWFs) explicitly and
using these for different purposes.  The reasons are manifold: First,
WFs, akin to LMO's in molecules,
provide an insightful chemical analysis of the nature of bonding,
and its evolution during, say, a chemical reaction.  As such, they
have become an established tool in the post-processing of
electronic-structure calculations. More interestingly, there are
formal connections between the centers of charge of the WFs
and the Berry phases of the Bloch functions as they are
carried around the Brillouin zone.  This connection is embodied in
the microscopic modern theory of polarization, alluded to above,
and has led to important advances in the characterization
and understanding of dielectric response and polarization in
materials.
Of broader interest to the entire condensed matter community is the
use of WFs in the construction of model Hamiltonians
for, e.g., correlated-electron and magnetic systems.
An alternative use of WFs as localized, transferable
building blocks has taken place in the theory of ballistic
(Landauer) transport, where Green's functions and self-energies can
be constructed effectively in a Wannier basis, or that of
first-principles tight-binding Hamiltonians, where chemically-accurate
Hamiltonians are constructed directly on the Wannier
basis, rather than fitted or inferred from macroscopic
considerations.
Finally, the
ideas that were developed for electronic WFs have
also seen application in very different contexts.  For
example, MLWF's have been used in
the theoretical analysis of phonons, photonic crystals, cold atom
lattices, and the local dielectric responses of insulators.

Here we review these developments.  We first introduce the transformation
from Bloch functions to WFs in Sec.~\ref{sec:theory},
discussing their gauge freedom and the methods developed for constructing WFs
through projection or maximal localization.  A
``disentangling procedure'' for constructing WFs
for a non-isolated set of bands (e.g., in metals) is also
described.  In Sec.~\ref{sec:locorb} we discuss variants of these
procedures in which different localization
criteria or different algorithms are used, and discuss the relationship
to ``downfolding'' and linear-scaling methods.
Sec.~\ref{sec:bonding} describes how the calculation of WFs
has proved to be a useful tool for analyzing the nature
of the chemical bonding in crystalline, amorphous, and defective systems.
Of particular importance is the ability to use WFs as
a local probe of electric polarization, as
described in Sec.~\ref{sec:pol}.  There we also discuss how the Wannier
representation has been useful in describing 
orbital magnetization, NMR chemical shifts, orbital magnetoelectric
responses, and topological insulators.  Sec.~\ref{sec:winterp}
describes Wannier interpolation schemes, by which quantities computed
on a relatively coarse $\kk$-space mesh can be used to interpolate
faithfully onto an arbitrarily fine $\kk$-space mesh at relatively
low cost.  In Sec.~\ref{sec:basis} we discuss applications in which
the WFs are used as an efficient basis 
for the calculations of quantum transport properties, the
derivation of semiempirical potentials, and for describing
strongly-correlated systems.  Sec.~\ref{sec:other} contains a
brief discussion of the construction and use of WFs
in contexts other than electronic-structure theory, including
for phonons in ordinary crystals, photonic crystals, and cold
atoms in optical lattices.  Finally, Sec.~\ref{sec:summary}
provides a short summary and conclusions.
