  I present results from an extended Migdal-Eliashberg theory of
electron-phonon interactions and superconductivity. The history of the
electron-phonon problem is introduced, and then study of the intermediate
parameter regime is justified from the energy scales in the cuprate
superconductors. The Holstein model is detailed, and limiting cases are
examined to demonstrate the need for an extended theory of superconductivity.
Results of the extended approximation are shown, including spectral functions
and phase diagrams. These are discussed with reference to Hohenberg's theorem,
the Bardeen-Cooper-Schrieffer theory and Coulomb repulsion.
