  This article gives a contemporary and to some extent pedagogical review of
the current theoretical understanding of the formation of the superconducting
state in metallic systems with a variable density of carriers. We make an
attempt to describe the crossover from the Bose-Einstein condensation type
(small densities) to the Bardeen-Cooper-Schrieffer one (large densities). The
functional methods are used throughout the treatment. The most of the results
are considered in a review form for the first time. Some of them (in
particular, the possible opening of a pseudogap) are used to explain the
experimental data avaliable for high-temperatures superconductors.
