The review deals with the main results of a consistent study of electron-lattice interaction in crystals in the strong-coupling limit. Small polaron and bipolaron formation is shown to provide a number of new physical phenomena both in the normal and superconducting states of the system. Two mechanisms of superconductivity are discussed in detail. The first one arises from the Cooper pairing of small polarons in momentum space (polaron superconductivity). The second one is due to polaron pairing in real space leading to on-site or intersite bipolaron formation with superconductivity analogous to the superfluidity of $^4$He (bipolaron superconductivity). Highly non-adiabatic motion of (bi)polarons in the narrow band results in fundamental differences of their superconducting state properties with respect to the predictions of the BCS theory and its well known strong-coupling generalization. A number of basic properties of high-temperature metallic oxides being analyzed in terms of (bi)polaron theory of superconductivity is shown to reveal a satisfactory agreement of its findings with the available experimental data.