  The physics of ferromagnetic doped manganites, such as
\chem{La_{1-x}Ca_xMnO_3} with $x\approx0.2$--0.4, is reviewed. The concept of
double exchange is discussed within the general framework of itinerant electron
magnetism. The new feature in this context is the coupling of electrons to
local phonon modes. Emphasis is placed on the quantum nature of the phonons and
the link with polaron physics. However it is stressed that the manganites fall
in an intermediate coupling regime where standard small-polaron theory does not
apply. The recently-developed many-body coherent potential approximation is
able to deal with this situation and Green's recent application to the Holstein
double-exchange model is described. Issues addressed include the nature of the
basic electronic structure, the metal-insulator transition, a unification of
colossal magnetoresistance, pressure effects and the isotope effect, pseudogaps
in spectroscopy and the effect of electron-phonon coupling on spin waves.
