The Morse decomposition theorem states that a compact invariant set of a given flow can be decomposed into finite invariant compact subsets and connecting orbits between them, which is helpful for us to study the inner structure of compact invariant sets. When dynamical systems are randomly perturbed, by real or white noise, what we concern is if we can study the structure of an invariant random compact sets, including global random attractors? We show that for finite and infinite dimensional random dynamical systems, we have the random Morse decomposition; we also construct Lyapunov function for the decomposition. By this, we may introduce the concept of random gradient system as a random semiflow possessing a continuous random Lyapunov function which describes the asymptotic behavior of the system. For deterministic systems, we introduce the concept of natural order to study the relative stability of Morse sets by the stochastic perturbation method. We also investigate the stochastic stability of Morse (invariant) sets under general white noise perturbations when the intensity of noise converges to zero.