The surface of one's face is a 2d manifold embedded in 3d space; the space of all faces and facial expressions is a manifold with perhaps a few hundred statistically significant dimensions. These are very useful manifolds. Inconveniently, we have access to these manifolds only through their unknown immersions in the much higher-dimensional space of all possible (retinal) images. This motivates the problem of characterizing a manifold from data — usually a sparse sample of discrete points, tangents, or local geodesic distances. I'll outline a remarkably straightforward method that uses nullspace computations to separate variation on the manifold from variation due to the immersion, and connect this solution to practical problems in data analysis, projective geometry, and graph embeddings. The talk will be peppered with examples from facial coding and synthesis, shape-from-silhouettes problems, acoustic modeling of vowels, and computer graphics.