The theory of nonlinear oscillations and motion stability is a fundamental part of the modern theory of differential equations and at the same time represents the basis for a number of important applied disciplines, in particular the theory of automatic control. The book at a high mathematical level in a clear and compact form presents the analytical apparatus for constructing solutions of linear, quasilinear and strongly nonlinear differential equations, as well as studying their stability. Particular attention is paid to ways of finding periodic and near-periodic solutions based on the ideas of Lyapunov, Poincaré, Krylov and Bogolyubov .