Stochastic differential equations play an increasingly important role in modeling the dynamics of a large variety of systems in the natural sciences, and in technological applications. This book is aimed at advanced undergraduate and graduate students, and researchers in mathematics, physics, the natural sciences, and engineering. It presents a new constructive approach to the quantitative description of solutions to systems of stochastic differential equations evolving on well-separated timescales. The method, which combines techniques from stochastic analysis and singular perturbation theory, allows the domains of concentration for typical sample paths to be determined, and provides precise estimates on the transition probabilities between these domains. In addition to the detailed presentation of the set-up and mathematical results, applications to problems in physics, biology, and climatology are discussed. The emphasis lies on noise-induced phenomena such as stochastic resonance, hysteresis, excitability, and the reduction of bifurcation delay. Nils Berglund joined the research group "Classical and Quantum Dynamics" at the Centre de Physique Théorique (CNRS) in Marseille-Luminy in 2001. He teaches in the Mathematics Department of the Université du Sud Toulon-Var. Barbara Gentz joined the research group "Interacting Random Systems" at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) in Berlin in 1998. She teaches in the Institute of Mathematics at the Technical University in Berlin. .