  Recent work on an electron-phonon model in two-dimensions is reviewed and
some new results are presented. We utilize both Quantum Monte Carlo simulations
and Migdal-Eliashberg theory. The type of electron-ion coupling considered is
on-site. Competing instabilities are charge density wave (CDW) and singlet
superconductivity. When the electron band is half-filled, the charge density
wave dominates. Away from half-filling CDW correlations actually suppress
superconductivity.
