The heat kernel expansion is a powerful tool for studying one-loop divergences, anomalies, and asymptotics of the effective action. This report collects useful information on heat kernel coefficients, providing explicit expressions for these coefficients on manifolds with and without boundaries, under various boundary conditions and in the presence of singularities. The coefficients are expressed in terms of geometric invariants, which are derived for scalar and spinor theories, Yang-Mills fields, gravity, and open bosonic strings. The report discusses the relationships between heat kernel coefficients and quantum anomalies, anomalous actions, and covariant perturbation expansions of the effective action. The heat kernel is a classical subject in mathematics, closely related to the eigenvalue asymptotics of differential operators. It is also an important tool in the study of the Atiyah-Singer index theorem. By the early 1990s, the heat kernel expansion on manifolds without or with boundaries and simple local boundary conditions was well understood. The heat kernel became a standard tool in calculations of the vacuum polarization, the Casimir effect, and in the study of quantum anomalies. Recent developments in theoretical physics and mathematics have led to a highly specialized field, with new results on non-standard boundary conditions and geometries. The report presents a unifying approach to the heat kernel expansion, providing a "user-friendly" guide to the field. The main idea is the universality of the heat kernel, which can be used in a wide variety of applications regardless of details such as spin, gauge group, etc. The heat kernel is particularly useful for studying quantum anomalies, various perturbative expansions of the effective action, and selected non-perturbative relations for the effective action. The report is organized into sections covering spectral functions, relevant operators and boundary conditions, heat kernel expansion on manifolds without and with boundaries, manifolds with singularities, anomalies, resummation of the heat kernel expansion, and exact results for the effective action. It includes detailed discussions on the heat kernel coefficients, their relation to the zeta function, and their use in regularizing the effective action. The report also addresses the limitations of the heat kernel formalism, particularly in the presence of spinorial background fields and beyond the one-loop approximation.The heat kernel expansion is a powerful tool for studying one-loop divergences, anomalies, and asymptotics of the effective action. This report collects useful information on heat kernel coefficients, providing explicit expressions for these coefficients on manifolds with and without boundaries, under various boundary conditions and in the presence of singularities. The coefficients are expressed in terms of geometric invariants, which are derived for scalar and spinor theories, Yang-Mills fields, gravity, and open bosonic strings. The report discusses the relationships between heat kernel coefficients and quantum anomalies, anomalous actions, and covariant perturbation expansions of the effective action. The heat kernel is a classical subject in mathematics, closely related to the eigenvalue asymptotics of differential operators. It is also an important tool in the study of the Atiyah-Singer index theorem. By the early 1990s, the heat kernel expansion on manifolds without or with boundaries and simple local boundary conditions was well understood. The heat kernel became a standard tool in calculations of the vacuum polarization, the Casimir effect, and in the study of quantum anomalies. Recent developments in theoretical physics and mathematics have led to a highly specialized field, with new results on non-standard boundary conditions and geometries. The report presents a unifying approach to the heat kernel expansion, providing a "user-friendly" guide to the field. The main idea is the universality of the heat kernel, which can be used in a wide variety of applications regardless of details such as spin, gauge group, etc. The heat kernel is particularly useful for studying quantum anomalies, various perturbative expansions of the effective action, and selected non-perturbative relations for the effective action. The report is organized into sections covering spectral functions, relevant operators and boundary conditions, heat kernel expansion on manifolds without and with boundaries, manifolds with singularities, anomalies, resummation of the heat kernel expansion, and exact results for the effective action. It includes detailed discussions on the heat kernel coefficients, their relation to the zeta function, and their use in regularizing the effective action. The report also addresses the limitations of the heat kernel formalism, particularly in the presence of spinorial background fields and beyond the one-loop approximation.