The heat kernel expansion is a powerful tool for studying one-loop divergences, anomalies, and various asymptotics of the effective action in quantum field theory. This report aims to compile useful information on heat kernel coefficients, which are scattered across mathematical and physical literature. The coefficients are presented for manifolds with and without boundaries, under local and non-local boundary conditions, and in the presence of various singularities such as domain walls. These coefficients are expressed in terms of geometric invariants derived for scalar and spinor theories with different interactions, 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, both at low and high energies. The content covers spectral functions, differential geometry, spectral geometry, and boundary conditions, providing a comprehensive guide to the heat kernel expansion.The heat kernel expansion is a powerful tool for studying one-loop divergences, anomalies, and various asymptotics of the effective action in quantum field theory. This report aims to compile useful information on heat kernel coefficients, which are scattered across mathematical and physical literature. The coefficients are presented for manifolds with and without boundaries, under local and non-local boundary conditions, and in the presence of various singularities such as domain walls. These coefficients are expressed in terms of geometric invariants derived for scalar and spinor theories with different interactions, 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, both at low and high energies. The content covers spectral functions, differential geometry, spectral geometry, and boundary conditions, providing a comprehensive guide to the heat kernel expansion.