We measured the ability to discriminate 3D shapes across changes in viewpoint and illumination based on rich monocular 3D information and tested whether the addition of stereo information improves shape constancy. Stimuli were images of smoothly curved, random 3D objects. Objects were presented in three viewing conditions that provided different 3D information: shading-only, stereo-only, and combined shading and stereo. Observers performed shape discrimination judgments for sequentially presented objects that differed in orientation by rotation of 0°-60° in depth. We found that rotation in depth markedly impaired discrimination performance in all viewing conditions, as evidenced by reduced sensitivity (d') and increased bias toward judging same shapes as different. We also observed a consistent benefit from stereo, both in conditions with and without change in viewpoint. Results were similar for objects with purely Lambertian reflectance and shiny objects with a large specular component. Our results demonstrate that shape perception for random 3D objects is highly viewpoint-dependent and that stereo improves shape discrimination even when rich monocular shape cues are available.