  Calculations of ground-state and excited-state properties of materials have
been one of the major goals of condensed matter physics. Ground-state
properties of solids have been extensively investigated for several decades
within the standard density functional theory. Excited state properties, on the
other hand, were relatively unexplored in ab initio calculations until a decade
ago. The most suitable approach up to now for studying excited-state properties
of extended systems is the Green function method. To calculate the Green
function one requires the self-energy operator which is non-local and energy
dependent. In this article we describe the GW approximation which has turned
out to be a fruitful approximation to the self-energy. The Green function
theory, numerical methods for carrying out the self-energy calculations,
simplified schemes, and applications to various systems are described.
Self-consistency issue and new developments beyond the GW approximation are
also discussed as well as the success and shortcomings of the GW approximation.
