The book discusses main forms of equations of motion of a rigid body, including the motion in potential fields, in fluid (Kirchhoffs equations), and motion of a rigid body with cavities filled with fluid. The book contains conditions of the order reduction of these equations, and existence of cyclic variables. It collects almost all integrable cases presently known, and methods of their explicit integration. For the purpose of investigation, the computer techniques, allowing vivid representation of the motion picture, are widely used. The majority of results presented in the book belongs to the authors.For students and graduate students of mechanical, mathematical and physical departments of universities, mathematical physicists, and specialists on dynamical systems.