One of the most enduring questions about human vision is how we are able to perceive three-dimensionality in two-dimensional images, even in the absence of motion, stereo, shading and texture cues. Traditionally, researchers have posited the use of innately specified brain mechanisms, such as a preference for simplicity. To test these ideas, we have developed a computational system for recovering 3D structures from single 2D line-drawings, using a fixed set of constraints that partly capture the notion of perceptual simplicity. While the system is able to mimic human performance for a small set of inputs, it exhibits significant limitations when analyzing natural imagery. To account for these shortcomings, we have proposed a learning-based theory, and have gathered experimental data that provide strong evidence for a role of object-specific learning in the perception of 3D structure. Together, the computational and experimental studies provide a good foundation for building a more comprehensive account of 3D shape perception in single 2D images.