The celebrated Lyapunov functionLyapunov function method is the main tool for stability analysis of nonlinear differential equations. The advantage of this method lies in the fact that a decision on the stability or instability of the system can be made by an investigation of the right-hand side of the differential equation without its finding a solution. Initially, the Lyapunov functionLyapunov function method was limited by a regular class of ODEsOrdinary differential equation (ODE) with continuous right-hand sides. The later evolution of the theory of differential equations and its applications had required extensions of this method to differential equations with discontinuous right-hand sides, functionalFunctional differential equations, PDEs and evolution equations in Banach spacesBanach space. This chapter surveys some elements of the Lyapunov functionsLyapunov function theory for finite-dimensional systems.