  We consider second-order elliptic partial differential operators acting on
sections of vector bundles over a compact Riemannian manifold without boundary,
working without the assumption of Laplace-like principal part $-\N^\mu\N_\mu$.
Our objective is to obtain information on the asymptotic expansions of the
corresponding resolvent and the heat kernel. The heat kernel and the Green's
function are constructed explicitly in the leading order. The first two
coefficients of the heat kernel asymptotic expansion are computed explicitly. A
new semi-classical ansatz as well as the complete recursion system for the heat
kernel of non-Laplace type operators is constructed. Some particular cases are
studied in more detail.
