The book describes the analytical and numerical methods of the theory of Hamiltonian systems and their applications to the study of movements close to the libration points of the organized problem of three bodies. The main attention is paid to the stability of equilibrium positions and periodic movements of nonlinear Hamiltonian systems in resonant cases, when purely imaginary characteristic indicators of the linearized system of equations of indignant motion are connected by integer relations .. The problem of the stability of triangular libration points of a limited task of three bodies has been studied in detail. A method for constructing and studying the stability of periodic movements close to the equilibrium positions of autonomous Hamiltonian systems has been developed. This method is used in the analysis of periodic movements close to triangular libration points. Due to the success in the exploration and use of outer space, long-known exact solutions to the classical problem of three bodies - libration points attract more and more attention. This book is devoted to a detailed study of the stability of triangular libration points of a limited problem of three bodies and elements of the theory of motion near libration points. An auxiliary, though significant, part of the book is a presentation of the theory of the stability of Hamiltonian systems. The book is based on a number of published works by the author. Some results from other authors have also been used. Many of the scientific issues raised in the book have been repeatedly discussed with B. A. Sarychev, V. In. Beletsky, A. D. Bruno . Live and friendly participation in . In. Beletsky contributed a lot to the very appearance of this book .. The author is very grateful.