The book begins with a review of the physical and mathematical motivations for studying physical theories in the presence of boundaries, with emphasis on electrostatics, vacuum Maxwell theory and quantum cosmology. The Feynman propagator is then studied in Minkowski space- time and in curved space-time. In the latter case, the corresponding Schwinger-DeWitt asymptotic expansion is given. The following chapters introduce the theory of the effective action and the geometric improvement due to Vilkovisky, the manifestly covariant quantization of gauge fields, zeta-function regularization in mathematics and in quantum field theory, and the choice of local or non-local boundary conditions in one-loop quantum theory. In the second part, the authors present their investigations of Euclidean Maxwell theory, simple supergravity and Euclidean quantum gravity. The recurring themes are the evaluation of conformal anomalies and one-loop divergences on manifolds with boundary, the relation between covariant and non-covariant formalisms for the quantization of gauge fields and the choice of boundary condition in the one-loop semiclassical analysis of Euclidean quantum gravity and simple supergravity. Audience: This work will appeal to research workers involved in quantum field theory, relativity and gravitation, electromagnetic theory, cosmology and quantum mechanics.