This talk is a theoretical, tutorial introduction to computational techniques for the representation and comparison of shapes, inspired by problems in medical imaging and gesture recognition. When defining distances between shapes, or average shapes, it is important to (i) establish point-wise correspondences between shapes and (ii) focus the attention of the metric on the scales of interest in the application. Both problems can be addressed through concepts based on the Laplacian of a function. Specifically, correspondence between two shapes can be established by creating a vector field in the volume that separates them. If this field satisfies the Laplace equation, then it generates geometrically smooth and well defined correspondences. Once a correspondence has been found, a point-wise measure of distance can be defined. The theory of the Laplace-Beltrami operator then yields methods for integrating this measure over the shapes in a way that emphasizes the scales of interest in a given application. All these concepts are traditionally defined in the continuous setting. For discrete, mesh-based shape representations, the boundary element method and techniques from spectral graph theory provide sound numerical techniques for implementation.