The heat kernel is one of the most powerful tools in modern theoretical and
mathematical physics, analysis on manifolds, and differential geometry. It is
associated with a second-order elliptic self-adjoint partial differential operator
of Laplace type acting on sections of a vector bundle over a Riemannian
manifold. In particular, the heat kernel gives the general framework for the
calculation of the effective action and correlation functions in quantum field
theory and statistical physics. Of special interest and great importance is the
asymptotic expansion of the heat kernel. The coefficients of the asymptotic
expansion of the trace of the heat kernel are spectral invariants of the
differential operator that describe the asymptotic properties of its spectrum.
They are of central interest in spectral geometry and are also closely allied
to the non-linear completely integrable systems, such as Korteweg-de Vries
hierarchy. In the situation when it is impossible to compute the trace of the
heat kernel exactly, it becomes very important to study various asymptotic
regimes and special cases.

We describe a general technique for calculation of the short time asymptotics
of the heat kernel and analyze the general structure of the heat invariants. 
They are local invariants built polynomially from the jets of the symbol of the
operator, i.e. from the curvatures and their covariant derivatives. It is known
that there is a certain grading associated with the number of derivatives of
the curvatures. We present the results of the calculation of the terms with 
leading derivatives which contain just five invariant structures. The opposite
case, i.e. the terms without any covariant derivatives, is much more
complicated. It is reduced to the calculation of the asymptotic expansion of the
heat kernel diagonal on symmetric spaces. We develop a new method for the
calculation of the heat kernel asymptotics that is based on a representation of
the heat semigroup in form of an average over the Lie group of isometries.
For the case of a scalar Laplacian we were able to carry out this program
and to obtain a generating function for the sequence of all heat kernel
coefficients on (locally) symmetric spaces.