Joint work with Mark Davenport, Marco Duarte, Chinmay Hegde, and Michael Wakin. Compressive sensing is a new approach to data acquisition in which sparse or compressible signals are digitized for processing not via uniform sampling but via measurements using more general, even random, test functions. In contrast with conventional wisdom, the new theory asserts that one can combine "low-rate sampling" (dimensionality reduction) with an optimization-based reconstruction process for efficient and stable signal acquisition. While the growing compressive sensing literature has focused on sparse or compressible signal models, in this talk, we will explore the use of manifold signal models. We will show that for signals that lie on a smooth manifold, the number of measurements required for a stable representation is proportional to the dimensionality of the manifold, and only logarithmic in the ambient dimension — just as for sparse signals. As an application, we consider learning and inference from manifold-modeled data, such as detecting tumors in medical images, classifying the type of vehicle in airborne surveillance, or estimating the trajectory of an object in a video sequence. Specific techniques we will overview include compressive approaches to the matched filter (dubbed the "smashed filter"), intrinsic dimension estimation for point clouds, and manifold learning algorithms. We will also present a new approach based on the joint articulation manifold (JAM) for compressive distributed learning, estimation, and classification.