For general images of smooth objects wholly contained in the field of view, and for illumination symmetric around the viewing direction, it is proven that shape is uniquely determined by shading. Thus, shape from shading is a well-posed problem under these illumination conditions; and regularization is unnecessary for surface reconstruction and should be avoided. Generic properties of surfaces and images are established. Questions of existence are also discussed. Under the conditions above, it is argued that most images are effectively impossible, with no corresponding physically reasonable surface, and that any image can be rendered effectively impossible by a small perturbation of its intensities. This is explicitly illustrated for a synthetic image. The proofs are based on ideas of dynamical systems theory and global analysis.