The book is devoted to some nonlocal problems of the theory of nonlinear oscillations. It examines non-autonomous systems of ordinary differential equations and examines various issues related to periodic and limited solutions. A significant part of the book is devoted to methods that have not yet been set forth in monographic literature. the method of guiding functions of proof of the existence of periodic and limited solutions , Research on Positive Periodic Decisions , elucidation of the links between the stability of periodic solutions and the concavity of the shift operator, application of cone theory to the study of periodic solutions, from the state of equilibrium and others. For the convenience of the reader, the book describes in detail the topological and functional-analytical concepts used in the study of periodic solutions of differential equations. The reader is required only to know the general facts of the theory of ordinary differential equations .. The book is designed for researchers, graduate students and students-mathematicians and mechanics interested in the theory of oscillations and qualitative theory of differential equations.