We review the status of covariant methods in quantum field theory and quantum
gravity, in particular, some recent progress in the calculation of the
effective action via the heat kernel method. We study the heat kernel
associated with an elliptic second-order partial differential operator of
Laplace type acting on smooth sections of a vector bundle over a Riemannian
manifold without boundary. We develop a manifestly covariant method for
computation of the heat kernel asymptotic expansion as well as new algebraic
methods for calculation of the heat kernel for covariantly constant background,
in particular, on homogeneous bundles over symmetric spaces, which enables one
to compute the low-energy non-perturbative effective action.Comment: 71 pages, 2 figures, submitted for publication in the Springer book
(in preparation) "Quantum Gravity", edited by B. Booss-Bavnbek, G. Esposito
and M. Lesc