Abstract We present a theory that addresses the problem of determining shape from the (small or differential) motion of an object with unknown isotropic re?ectance, under arbitrary unknown distant illumination, for both orthographic and perpsective projection. Our theory imposes fundamental limits on the hardness of surface reconstruction, independent of the method involved. Under orthographic projection, we prove that three differential motions suf?ce to yield an invariant that relates shape to image derivatives, regardless of BRDF and illumination. Under perspective projection, we show that four differential motions suf?ce to yield depth and a linear constraint on the surface gradient, with unknown BRDF and lighting. Further, we delineate the topological classes up to which reconstruction may be achieved using the invariants. Finally, we derive a general strati?cation that relates hardness of shape recovery to scene complexity. Qualitatively, ou invariants are homogeneous partial differential equations for simple lighting and inhomogeneous for complex illumination. Quantitatively, our framework shows that the minimal number of motions required to resolve shape is greater for more complex scenes. Prior works that assume brightness constancy, Lambertian BRDF or a known directional light source follow as special cases of our strati?cation. We illustrate with synthetic and real data how potential reconstruction methods may exploit our framework.