  This paper is an overview on our recent results in the calculation of the
heat kernel in quantum field theory and quantum gravity. We introduce a
deformation of the background fields (including the metric of a curved
spacetime manifold) and study various asymptotic expansions of the heat kernel
diagonal associated with this deformation. Especial attention is payed to the
low-energy approximation corresponding to the strong slowly varying background
fields. We develop a new covariant purely algebraic approach for calculating
the heat kernel diagonal in low-energy approximation by taking into account a
finite number of low-order covariant derivatives of the background fields, and
neglecting all covariant derivatives of higher orders. Then there exist a set
of covariant differential operators that together with the background fields
and their low-order derivatives generate a finite dimensional Lie algebra. In
the zeroth order of the low-energy perturbation theory, determined by
covariantly constant background, we use this algebraic structure to present the
heat operator in the form of an average over the corresponding Lie group. This
simplifies considerably the calculations and allows to obtain closed explicitly
covariant formulas for the heat kernel diagonal. These formulas serve as the
generating functions for the whole sequence of the Hadamard-Minakshisundaram-
De Witt-Seeley coefficients in the low-energy approximation.
