The interplay between functional and stochastic analysis has wide implications for problems in partial differential equations, noncommutative or "free" probability, and Riemannian geometry. Written by active researchers, each of the six independent chapters in this volume is devoted to a particular application of functional analytic methods in stochastic analysis, ranging from work in hypoelliptic operators to quantum field theory. Every chapter contains substantial new results as well as a clear, unified account of the existing theory; relevant references and numerous open problems are also included. Key topics: * Stochastic differential equations (SDEs), hypoelliptic operators, and SDEs based on Levy processes * Stochastic calculus on Riemannian manifolds and curved Wiener spaces * Noncommutative and quantum probability * The Feynman integral, evolution processes, the Feynman--Kac formula, and applications to quantum field theory * Convolution operators and the amenability of the underlying locally compact groups, with connections among classical random walks, spectral theory, and Beurling and Segal subalgebras Self-contained, well-motivated, and replete with suggestions for further investigation, this book will be especially valuable as a seminar text for dissertation-level graduate students. Research mathematicians and physicists will also find it a useful and stimulating reference. Contributors: D.R. Bell; B.K. Driver; S. Gudder; B. Jefferies; H. Kunita; and M.M. Rao.