Lyapunov exponents and stabiliy The monograph contains the necessary information from the modern theory of Lyapunov characteristic exponents of ordinary linear differential systems. It is mainly dedicated to the brief description of the results obtained by the author connected with the development of the following parts: the theory of Perron lower exponents the freezing method the theory of exponential and sigma exponents and their connection with characteristic central and general exponents the dependence of characteristic exponents of linear systems on exponentially decreasing perturbation and the theory of their stability with respect to small perturbations. As an application of those results the author considered the Lyapunov problem on the exponential stability of an ordinary differential system by linear approximation. In the monograph the method of rotations by V.M.Millionschikov is systematically used. This volume is intended for specialists in the asymptotic theory of ordinary differential systems and the stability theory for post graduates and students specialized in the field of differential equations.