  Eliashberg theory is a theory of superconductivity that describes the role of
phonons in providing the attractive interaction between two electrons. Phonon
dynamics are taken into account, thus giving rise to retardation effects that
impact the electrons, in the form of a frequency-dependent electron
self-energy. In the superconducting state, this means that the order parameter,
generally considered to be a static quantity in the Bardeen-Cooper-Schrieffer
(BCS) theory, also becomes frequency-dependent. Here we review the finite
temperature formulation of Eliashberg theory, both on the imaginary and real
frequency axis, and briefly display some examples of the consequences of a
dynamical, as opposed to static, interaction. Along the way we point out where
further work is required, concerning the validity of some of the assumptions
used.
