  An introduction is given to the many-body coherent potential approximation
(CPA) for the double-exchange (DE) model and the Holstein-DE model, the latter
including coupling of the electrons to local phonons as well as to local spins.
It is shown how the method can treat the local spins and phonons
quantum-mechanically and how it is equivalent to dynamical mean field theory in
the classical limit. In the Holstein-DE model a full discussion is given of the
cross-over from weak electron-phonon coupling through intermediate coupling,
where small-polaron bands begin to appear, to strong coupling where some
results similar to those of standard small-polaron theory are recovered. The
theory is applied to ferromagnetic manganites with a full discussion of
magnetic, transport and spectroscopic data. It is found that many manganites
are in the critical regime on the verge of small-polaron formation, which
explains their sensitivity to parameters such as applied magnetic field and
pressure.
