  This article reviews the theory of electron-phonon interactions in solids
from the point of view of ab-initio calculations. While the electron-phonon
interaction has been studied for almost a century, predictive non-empirical
calculations have become feasible only during the past two decades. Today it is
possible to calculate from first principles many materials properties related
to the electron-phonon interaction, including the critical temperature of
conventional superconductors, the carrier mobility in semiconductors, the
temperature dependence of optical spectra in direct and indirect-gap
semiconductors, the relaxation rates of photoexcited carriers, the electron
mass renormalization in angle-resolved photoelectron spectra, and the
non-adiabatic corrections to phonon dispersion relations. Here we review the
theoretical and computational framework underlying modern electron-phonon
calculations from first principles, as well as landmark investigations of the
electron-phonon interaction in real materials. In the first part of the article
we summarize the elementary theory of electron-phonon interactions and their
calculations based on density-functional theory. In the second part we discuss
a general field-theoretic formulation of the electron-phonon problem, and
establish the connection with practical first-principles calculations. In the
third part we review a number of recent investigations of electron-phonon
interactions in the areas of vibrational spectroscopy, photoelectron
spectroscopy, optical spectroscopy, transport, and superconductivity.
