The celebrated Lyapunov function method (or the direct Lyapunov method) introduced in the Ph.D. thesis of A. M. Lyapunov in 1892 is a simple effective tool for stability analysis of differential equations. The main advantage of this method lies in the fact that a decision on stability or instability can be made by means of a certain investigation of the right-hand side of a differential equation without finding its solutions. Initially, the Lyapunov function method was limited by a regular class of ODE with continuous right-hand sides. The later evolution of the ODE theory and its applications had required extensions of this method to differential equations with discontinuous right-hand sides, functional differential equations, PDEs, and evolution systems.