Measuring the similarity between 3D shapes is a fundamental problem, with applications in computer vision, molecular biology, computer graphics, and a variety of other fields. A challenging aspect of this problem is to find a suitable shape signature that can be constructed and compared quickly, while still discriminating between similar and dissimilar shapes. In this paper, we propose and analyze a method for computing shape signatures for arbitrary (possibly degenerate) 3D polygonal models. The key idea is to represent the signature of an object as a shape distribution sampled from a shape function measuring the global geometric properties of an object. The primary motivation for this approach is to reduce the shape matching problem to the comparison of probability distributions, which is simpler than traditional shape matching methods that require pose registration, feature correspondence or model fitting. We find that the dissimilarities between sampled distributions of simple shape functions (e.g. the distance between two random points on a surface) provide a robust method for discriminating between classes of objects (e.g. cars versus airplanes) in a moderately sized database, despite the presence of arbitrary translations, rotations, scales, reflections, tessellations, simplifications and model degeneracies. They can be evaluated quickly, and thus the proposed method could be applied as a pre-classifier in an object recognition system or in an interactive content-based retrieval application.