This book is aimed at theoretical and mathematical physicists and mathematicians interested in modern gravitational physics I have thus tried to use language familiar to readers working on classical and quantum gravity, paying attention both to difficult calculations and to existence theorems, and discussing in detail the current literature The first aim of the book is to describe recent work on the problem of boundary conditions in one loop quantum cosmology The motivation of this research was to under stand whether supersymmetric theories are one loop finite in the presence of boundaries, with application to the boundary value problemsoccurring in quantum cosmology Indeed, higher loop calculations in the absence of boundaries are already available in the litera ture, showing that supergravity is not finite I believe, however, that one loop calculations in the presence of boundaries arefundamental, in that they provide adirect check of the inconsistency of supersymmetric quantum cosmology from the perturbative point of view It therefore appears that higher order calculations are not strictly needed, if the one loop test already yields negative results Even though the question is not yet settled, this research has led to many interesting, new applications of areas of theoretical and mathematical physics such as twistor theory in flat space, self adjointness theory, the generalized Riemann zeta function, and the theory of boundary counterterms in super gravity I have also compared in detail my work with results by other authors, explaining, whenever possible, the origin of different results, the limits of my work and the unsolved problems