This volume presents stability theory for ordinary differential equations, discrete systems and systems on time scale, functional differential equations and uncertain systems via multicomponent Liapunov's functions. The book sets out a new approach to solution of the problem of constructing Liapunov's functions for three classes of systems of equations. This approach is based on the application of matrix-valued function as an appropriate tool for scalar or vector Liapunov function. The volume proposes an efficient solution to the problem of robust stability of linear systems. In terms of hierarchical Liapunov function the dynamics of neural discrete-time systems is studied and includes the case of perturbed equilibrium state.