  The study of spectral properties of natural geometric elliptic partial
differential operators acting on smooth sections of vector bundles over
Riemannian manifolds is a central theme in global analysis, differential
geometry and mathematical physics. Instead of studying the spectrum of a
differential operator $L$ directly one usually studies its spectral functions,
that is, spectral traces of some functions of the operator, such as the
spectral zeta function $\zeta(s)=\Tr L^{-s}$ and the heat trace
$\Theta(t)=\Tr\exp(-tL)$. The kernel $U(t;x,x')$ of the heat semigroup
$\exp(-tL)$, called the heat kernel, plays a major role in quantum field theory
and quantum gravity, index theorems, non-commutative geometry, integrable
systems and financial mathematics. We review some recent progress in the study
of spectral asymptotics. We study more general spectral functions, such as $\Tr
f(tL)$, that we call quantum heat traces. Also, we define new invariants of
differential operators that depend not only on the their eigenvalues but also
on the eigenfunctions, and, therefore, contain much more information about the
geometry of the manifold. Furthermore, we study some new invariants, such as
$\Tr\exp(-tL_+)\exp(-sL_-)$, that contain relative spectral information of two
differential operators. Finally we show how the convolution of the semigroups
of two different operators can be computed by using purely algebraic methods.
