The principal purpose of this book is to provide an account of the circle of ideas, results and techniques, which emerged roughly over the last ten years in the study of Brownian motion and random obstacles. The accumulation of results in many separate sources eventually made it impractical, if not impossible, for the nonspecialist to gain access to the developments of the subject. This book is an attempt to remedy this situation. Part of the thrill of the investigation of Brownian motion and random obsta cles certainly stems from its many connections with various areas of math ematics, but also from the formal and mysterious physical heuristics which relate to it. In particular the loose concept of pockets of low local eigenval ues plays an important role in the study of Brownian motion and random obstacles, and also represents a paradigm which has natural resonances with several other areas of random media. This last feature has increasingly be come clear over the last few years.