A large number of papers written over the last ten years have concerned the spectral theory of Laplace–Beltrami operators on complete Riemannian manifolds, and of other self-adjoint second order elliptic operators. Much of the interest has centred on the relationship between various types of Sobolev inequality, parabolic Harnack inequalities and the Liouville property on the one hand, and Gaussian heat kernel bounds on the other. For manifolds of bounded geometry there is an important connection between this problem and a corresponding one for discrete Laplacians on graphs. Standard references are [9, 37] and more recent literature can be traced via [5, 16, 32].