Plenary lecture is devoted to fundamental and frontier problems on stability and stabilization of control systems.Classical examples of non-stationary stabilization of linear systems are considered.Two approaches to the solution of the Brockett problem on non-stationary stabilization are presented [1-2].Problems of delayed feedback stabilization of unstable equilibria and unstable periodic orbits are discussed [3].An effective analytical-numerical method for the localization of undesired hidden oscillations and the construction of counterexamples to famous Aizermans and Kalmans conjectures on absolute stability of nonlinear control system is presented [4-13].An experiment by I.A.Blekhman,confirming the theory of elastic system stabilization developed by academician V.N.Chelomey,is presented [14].The problems of drill string failures [15-16],design of aircraft control systems and anti-windup schemes are discussed [17-19].Effective methods for nonlinear analysis and design of phase-locked loop based systems are demonstrated [20-24].