Some problems of dynamics and stability of motion of rigid body and systems consisting of rigid, elastic bodies or bodies with fluid - containing cavities are discussed. It begins with a brief account of the theory of integrability of ordinary differential equa tions and formulation of main theorems on stability and instability of motion of a system with finite degrees of freedom as well as the system with parameter distribution and described by joint system of ordinary and partial differential equations. Further the problems of dynamics and stability of steady motions of rigid body, gyrostats of different types and also the stability of steady motions of satellites with spinning rotors and having fluid-containing cavities and elastic parts are successively considered.