This paper presents a simple, flexible, and efficient algorithm for constructing a possibly non-convex, simple polygon that characterizes the shape of a set of input points in the plane, termed a characteristic shape. The algorithm is based on the Delaunay triangulation of the points. The shape produced by the algorithm is controlled by a single normalized parameter, which can be used to generate a finite, totally ordered family of related characteristic shapes, varying between the convex hull at one extreme and a uniquely defined shape with minimum area. An optimal O(nlogn) algorithm for computing the shapes is presented. Characteristic shapes possess a number of desirable properties, and the paper includes an empirical investigation of the shapes produced by the algorithm. This investigation provides experimental evidence that with appropriate parameterization the algorithm is able to accurately characterize the shape of a wide range of different point distributions and densities. The experiments detail the effects of changing parameter values and provide an indication of some “good” parameter values to use in certain circumstances.