Studies in the field of superconductivity theory is one of the most bright, fruitful, and promising trends in the theoretical physics of condensed matter, since superconductivity remains to be one of the most interesting research areas in physics. In this review consider such topics of high-T c superconductivity as the structures of high-T c superconductors, phase diagrams, and the problem of pseudogaps and analyze the mechanisms of superconductivity. We present general arguments as for the pairing symmetry in cuprate superconductors and investigate their thermodynamical properties within the spin-fluctuation mechanism of superconductivity, by using the method of functionals. This review deals with a wide scope of theoretical and experimental topics in superconductivity.