The phenomenon of neutrino oscillations is reviewed. The case of mixing of two neutrinos (both Majorana and Dirac neutrinos) of masses $m_1$ and $m_2$ is considered in detail. It is shown that the mixing hypothesis is consistent with the existing data if $\vert m_1^2-m_2^2\vert\lesssim$ 1eV$^2$ (for maximal mixing). A discussion is given of possible experiments to search for oscillations of neutrinos from reactors, meson factories, and highenergy accelerators. Oscillations can be observed in these experiments if $\vert m_1^2-m_2^2\vert\gtrsim10^{-2}$eV$^2$. The prospects of searches for neutrino oscillations in experiments using cosmic neutrinos, and in particular solar neutrinos, are discussed in detail. Such experiments have a unique sensitivity as tests of the mixing hypothesis (in the case of solar neutrinos, mixing effects would be observed if $\vert m_1^2-m_2^2\vert\gtrsim10^{-12}$eV$^2$. The ``solar neutrino paradox'' is discussed in detail from the point of view of neutrino mixing. Neutrino oscillations are also considered in the case when there exist in nature $N$>2 types of neutrinos. Schemes involving the mixing of heavy leptons are discussed. It is shown in the framework of a specific scheme involving right-handed currents that for heavy-lepton masses of the order of several GeV the probabilities of processes such as $\mu\to e\gamma$ and $\mu\to3e$ can be close to the existing experimental upper limits. The probabilities of these processes are practically zero when only the neutrinos are mixed.