  An overview about recent progress in the calculation of the heat kernel and
the one-loop effective action in quantum gravity and gauge theories is given.
We analyse the general structure of the standard Schwinger-De Witt asymptotic
expansion and discuss the applicability of that to the case of strongly curved
manifolds and strong background fields. We argue that the low-energy limit in
gauge theories and quantum gravity, when formulated in a covariant way, should
be related to background fields with covariantly constant curvature, gauge
field strength and potential term. It is shown that the condition of the
covariant constancy of the background curvatures brings into existence some Lie
algebra. The heat kernel operator for the Laplace operator is presented then as
an average over the corresponding Lie group with some nontrivial Gaussian
measure. Using this representation the heat kernel diagonal is obtained. The
result is expressed purely in terms of curvature invariants and is explicitly
covariant. Related topics concerning the structure of symmetric spaces and the
calculation of the effective action are discussed.
