Over the last several years, the author and his colleagues have developed new methods for the exact analysis, including numerical evaluations, of different spectral functions that have significant applications in physics. This book provides a comprehensive description of these techniques and of their various applications. The approach allows for an exact treatment of zeta functions associated with the spectrum of a certain class of second order elliptic differential operators. Although much of the book is devoted to spectral analysis and spectral geometry, it also demonstrates some of the many ramifications spectral analysis and geometry has in physics, particularly to heat asymptotics, quantum field theory, statistical mechanics, the theory of partitions, and Bose-Einstein condensation.