This paper is dedicated to the study of the problem of asymptotic stability for general non-autonomous dynamical systems (both with continuous and discrete time). We study the relation between diferent types of attractions and asymptotic stability in the framework of general non-autonomous dynamical systems. Specially we investigate the case of almost periodic systems, i.e., when the base (driving system) is almost periodic. We apply the obtained results we apply to diferent classes of non-autonomous evolution equations: Ordinary Diferential Equations, Functional Diferential Equations (both with finite retard and neutral type) and Semi-Linear Parabolic Equations.