  We give a short overview of the effective action approach in quantum field
theory and quantum gravity and describe various methods for calculation of the
asymptotic expansion of the heat kernel for second-order elliptic partial
differential operators acting on sections of vector bundles over a compact
Riemannian manifold. We consider both Laplace type operators and non-Laplace
type operators on manifolds without boundary as well as Laplace type operators
on manifolds with boundary with oblique and non-smooth boundary conditions.
