  The Hubbard-Holstein model is a simple model including both electron-phonon
interaction and electron-electron correlations. We review a body of theoretical
work investigating the effects of strong correlations on the electron-phonon
interaction. We focus on the regime, relevant to high-T_c superconductors, in
which the electron correlations are dominant. We find that the electron-phonon
interaction can still have important signatures, even if many anomalies appear,
and the overall effect is far from conventional. In particular in the
paramagnetic phase the effects of phonons are much reduced in the low-energy
properties, while the high-energy physics can be strongly affected by phonons.
Moreover, the electron-phonon interaction can still give rise to important
effects, like phase separation and charge-ordering, and it assumes a
predominance of forward scattering even if the bare interaction is assumed to
be local (momentum independent). Antiferromagnetic correlations reduce the
screening effects due to electron-electron interactions and revive the
electron-phonon effects.
