% \section{SUMMARY AND PROSPECTS}
% \label{sec:summary}

In this review, we have summarized methods for constructing
WFs to represent electrons in periodic solids
or other extended systems.  While several methods
have been surveyed, our emphasis has been on the one of
\textcite{marzari-prb97}, essentially the generalization
of the approach of Foster and Boys
\cite{boys-rmp60,foster-rmp60a,foster-rmp60b,boys-66} to
periodic systems, in which the gauge freedom is resolved
by minimizing the sum of the quadratic spreads of the WFs.
The widespread adoption of this approach is reflected in the fact
that it has been incorporated as a feature into a large number
of modern first-principles electronic-structure code packages
including
{\sc Quantum ESPRESSO} \cite{gianozzi-jpcm09},
{\sc ABINIT} \cite{gonze-cpc09},
{\sc FLEUR} \cite{freimuth-prb08},
{\sc WIEN2k} \cite{schwarz-cpc02,kunes-cpc10},
{\sc Siesta} \cite{soler-jpcm02},
and {\sc VASP} \cite{kresse-prb96}.
In the above cases
this has been done by providing an interface to the Wannier90
package \cite{mostofi-cpc08}, an open-source post-processing
code developed by the Authors, offering most of the capabilities described 
in this review. Other efforts have also seen
the implementation of MLWFs in CPMD \cite{cpmd}, 
GPAW \cite{enkovaara-jpcm10}, OpenMX \cite{openmx}, and 
WanT \cite{calzolari-prb04,want} - this latter, and Wannier90, also allowing for quantum-transport calculations.

After an initial wave of applications increased the visibility of
WFs in the community and demonstrated their utility for a variety
of applications, other methods for constructing WFs were also
developed, as discussed in Secs.~\ref{sec:theory} and
\ref{sec:locorb}.  For some purposes, e.g., for many plane-wave
based LDA+U and DMFT calculations, methods based on simple
projection onto trial orbitals proved sufficient.  Methods tuned
specifically to $\Gamma$-point sampling of the BZ for supercell
calculations also became popular.  And, as surveyed briefly
in Sec.~\ref{sec:other}, the construction and application of WFs
was also extended to periodic systems outside the electronic-structure
context, e.g., to phonons, photonic crystals, and cold-atom optical
lattices.

Still, the vast majority of applications of WF methods have been
to electronic structure problems.  The breadth of such applications
can be appreciated by reviewing the topics covered in
Secs.~\ref{sec:bonding}-\ref{sec:basis}.  Very broadly, these
fall into three categories: investigations into the nature of chemical
bonding (and, in complex systems such as liquids, the statistics
of chemical bonding), as discussed in Sec.~\ref{sec:bonding};
applications that take advantage of the natural ability of
WFs and WF charge centers to describe dipolar and orbital
magnetization phenomena in dielectric, ferroelectric, magnetic,
and magnetoelectric materials, as reviewed in
Sec.~\ref{sec:pol}; and the use of WF for basis functions, as
surveyed in Secs.~\ref{sec:winterp}-\ref{sec:basis}.

Today these methods find applications in many topical areas
including investigations into novel superconductors, multiferroics,
and topological insulators.
The importance of WFs is likely to
grow in response to future trends in computing, which are clearly
moving in the direction of more massive parallelization based on
algorithms that can take advantage of real-space partitioning.
This feature of WFs should also facilitate their adoption in
formulating new beyond-DFT methods in which many-body interactions
are included in a real-space framework.  Thus, the growing pressures
for increased efficiency and accuracy are likely to elevate the
importance of WF-based methods in coming years.
Overall, one can look forward
to continued innovation in the development and application of
WF-based methods to a wide variety of problems in modern
condensed-matter theory.
