The present paper is a survey of the contemporary state of art in the theory of multivalued mappings. In it one considers different forms of continuity of multivalued mappings, one investigates differentiable and measurable multivalued mappings, one considers single-valued continuous approximations and sections of multivalued mappings, one studies fixed points of multivalued mappings and other questions of this theory. One gives references to the literature regarding applications to the theory of games, mathematical economics, the theory of differential inclusions and generalized dynamical systems. The paper contains an extensive bibliography.