  We present a short review of our basic understanding of the physics of copper
- oxide superconductors and formulate the list of "solved" and "unsolved"
problems. The main problem remains theoretical description of the properties of
the normal state, requiring clarification of the nature of the so called
pseudogap state. We review simplified models of the pseudogap state, based on
the scenario of strong electron scattering by (pseudogap) fluctuations of
"dielectric" (AFM, CDW) short - range order and the concept of "hot" spots
(patches) on the Fermi surface. Pseudogap fluctuations are described as
appropriate static Gaussian random field scattering electrons. We derive the
system of recurrence equations for the one - particle Green's function and
vertex parts, taking into account all Feynman diagrams for electron scattering
by pseudogap fluctuations. Results of calculations of spectral density, density
of states and optical conductivity are presented, demonstrating both pseudogap
and localization effects. We analyze the anomalies of superconducting state
(both s- and d - wave pairing) forming on the "background" of these pseudogap
fluctuations. Microscopic derivation of Ginzburg - Landau expansion allows
calculations of critical temperature T_c and other basic characteristics of a
superconductor, depending on the parameters of the pseudogap. We also analyze
the role of "normal" (nonmagnetic) impurity scattering. It is shown that our
simplified model allows semiquantitative modelling of the typical phase diagram
of superconducting cuprates.
