We present a review of theoretical and experimental works on the problem of mutual interplay of Anderson localization and superconductivity in strongly disordered systems. We start with brief discussion of modern aspects of localization theory including the basic concept of scaling, self-consistent theory and interaction effects. After that we analyze disorder effects on Cooper pairing and superconducting transition temperature as well as Ginzburg-Landau equations for superconductors which are close to the Anderson transition. A necessary generalization of usual theory of ``dirty'' superconductors is formulated which allows to analyze anomalies of the main superconducting properties close to disorder-induced metal-insulator transition. Under very rigid conditions superconductivity may persist even in the localized phase (Anderson insulator).Strong disordering leads to considerable reduction of superconducting transition temperature T_{c} and to important anomalies in the behavior of the upper critical field H_{c2}. Fluctuation effects are also discussed. In the vicinity of Anderson transition inhomogeneous superconductivity appears due to statistical fluctuations of the local density of states. We briefly discuss a number of experiments demonstrating superconductivity close to the Anderson transition both in traditional and high-T_{c} superconductors. In traditional systems superconductivity is in most cases destroyed before metal-insulator transition. In case of high-T_{c} superconductors a number of anomalies show that superconductivity is apparently conserved in localized phase before it is suppressed by strong enough disorder.

120 pages, 41 figures, RevTeX 3.0, Published in Physics Reports 282,No.5&6,225 (1997), figures added