The purpose of this article is to establish upper and lower estimates for the integral kernel of the
semigroup exp(−t P) associated with a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to t ∈ C+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.