This volume is concerned with various kinds of limit theorems for stochastic processes defined as a result of random perturbations of dynamical systems; especially with the long-time behavior of the perturbed system. In particular, exit problems, metastable states, optimal stabilization, and asymptotics of stationary distributions are also carefully considered. The authors' main tools are the large deviation theory, the central limit theorem for stochastic processes, and the averaging principle--all presented in great detail. The results allow for explicit calculations of the asymptotics of many interesting characteristics of the perturbed system. Most of the results are closely connected with PDE's and the author's approach presents a powerful method for studying the asymptotic behavior of the solutions of initial-boundary value problems for corresponding PDE's. The most essential additions and changes in this new edition concern the averaging principle. A new chapter on random perturbations of Hamiltonian systems has been added along with new results on fast oscillating perturbations of systems with conservation laws. New sections on wave front propagation in semilinear PDE's and on random perturbations of certain infinite-dimensional dynamical systems have been incorporated into the chapter on Sharpenings and Generalizations.