The basic concepts and theorems of modern stability theory for systems in the first approximation, conservative systems, systems taking into account dissipative and gyroscopic forces, systems with a small parameter are presented. The focus is on Lyapunov's second method and its modifications. The theoretical material in the manual is accompanied by various examples of the application of stability theory in aviation and astronautics. Interesting dynamic phenomena are considered: beating, self-oscillation, flutter, external stability of resonances.