A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. These include not only the classical heat trace asymptotics and heat content asymptotics, but also the more exotic objects encountered in the context of manifolds with boundaries and imposing suitable boundary conditions. To date, however, there has been no unified discussion of these results.Asymptotic Formulae in Spectral Geometry collects these results and computations into one book. The author focuses on the functorial and special cases methods of computing asymptotic heat trace and heat content coefficients in the heat equation and introduces results derived from the Seeley Calculus and other methods. He incorporates the work of many authors into the presentation, and includes a complete bibliography that serves as a roadmap to the literature on the subject.The formulas studied here are important not only for their intrinsic interest, but also for their applications to areas such as index theory, compactness theorems for moduli spaces of isospectral metics, and zeta function regularization. Geometers, mathematical physicists, and analysts alike will undoubtedly find this book to be up to date, well organized, and broad in scope-in short, the definitive book on the subject.