  I review the quantum theory of the electron moving in a random environment.
First, the quantum mechanics of individual particles scattered on a random
potential is discussed. The quantum-mechanical description is extended to
many-body systems by using many-body Green functions. The many-body approach is
used to derive the coherent-potential approximation and to show how it fits
into the dynamical mean-field theory. The generating functional of the
coherent-potential approximation is obtained in an analytic form from the limit
to infinite dimensions of the general many-body description of non-interacting
electrons in random lattices. The analytic generating functional of the
mean-field description of random systems is extended to the Falicov-Kimball
model with thermally equilibrated scattering lattice potential. The many-body
Green functions are then used to calculate transport properties. The electrical
conductivity of the coherent-potential approximation is derived from the
two-particle Green function calculated in infinite spatial dimensions. Finally,
a perturbation theory for the vertex corrections to the mean-field conductivity
is introduced. In particular, it is shown how to make the expansion beyond the
local approximations conserving.
