If the neutrino has mass and exhibits mixing, this must necessarily result in oscillations, i.e., periodic transformation of one type of neutrino into another and vice versa. In general, these oscillations depend on the properties of the medium in which the neutrinos propagate. The propagation of a neutrino through a medium of varying density is, in general, accompanied by a number of new oscillation phenomena that are resonant in character. These phenomena have analogs in different branches of physics, including mechanics, because the neutrino oscillations constitute an oscillatory process that is unrelated to the quantum nature of particles. This review discusses practically all the possible aspects of the effect of a medium on neutrino oscillations. The Introduction presents general ideas on neutrino oscillations and on the mechanism responsible for the effect of the medium on the oscillations. It also discusses the equations describing the evolution of neutrinos in matter. This is followed by an account of the theory of neutrino oscillations in media with different density distributions. Particular attention is devoted to the most interesting case of a slow variation in density (the adiabatic case). This theory is a direct generalization of the theory of vacuum oscillations, originally developed by B. M. Pontecorvo. The separation of wave packets and the absorption of neutrinos in the medium are taken into account in the next Section. The oscillations of three types of neutrino in the most general case of mass hierarchy are also discussed. The last Section examines possible manifestations of resonant oscillations during the passage of neutrinos through the Sun and the Earth.