Spectral geometry with operators of Laplace type is introduced by discussing the inverse problem in the theory of vibrating membranes. This means that a spectrum of eigenvalues is given, and one would like to determine uniquely the shape of the vibrating object from the asymptotic expansion of the integrated heat kernel. It turns out that, in general, it is not possible to tell whether the membrane is convex, or smooth, or simply connected, but results of a limited nature can be obtained. These determine, for example, the volume and the surface area of the body. Starting from these examples, the very existence of the asymptotic expansion of the integrated heat hernel is discussed, relying on the seminal paper by Greiner (1971). A more careful analysis of the boundary-value problem is then performed, and the recent results on the asymptotics of the Laplacian on a manifold with boundary are presented in detail. For this purpose, one studies second-order elliptic operators with leading symbol given by the metric. The behaviour of the differential operator, boundary operator and heat-kernel coefficients under conformal rescalings of the background metric leads to a set of algebraic equations which, jointly with some results on product manifolds, determine a number of coefficients in the heat-kernel asymptotics. Such a property holds whenever one studies boundary conditions of Dirichlet or Robin type, or a mixture of the two. The heat-equation approach to index theorems, and the link between heat equation and ζ-function, are then described. The chapter ends with an introduction to the method used by McKean and Singer in their analysis of heat-kernel asymptotics.