The new phenomenon of twinning plane superconductivity is reviewed. It has already been observed in several metals. The superconductivity is localized near a twinning plane and it appears at a temperature higher than the critical temperature of the superconducting transition of a bulk metal. The phase diagram for the twinning-plane superconductivity, plotted using the magnetic field and temperature as the coordinates, is very different for type I and type II superconductors. A detailed analysis of the experimental data on the twinning-plane superconductivity is accompanied by a theoretical description of the phenomenon. The presence of a dense twinned structure may increase considerably the critical temperature: for example, in the case of tin it has been possible to increase this temperature more than threefold. We shall discuss also the data showing that the twinning-plane superconductivity can play an important role in the recently discovered high-temperature superconductors.