A laminar cortical model of stereopsis and later stages of 3D surface perception is developed and simulated. The model describes how initial stages of monocular and binocular oriented filtering interact with later stages of 3D boundary formation and surface filling-in in the lateral geniculate nucleus and cortical areas V1, V2, and V4. In particular, it details how interactions between layers 4, 3B, and 2/3A in V1 and V2 contribute to stereopsis, and clarifies how binocular and monocular information combine to form 3D boundary and surface representations. Along the way, the model modifies and significantly extends the disparity energy model. Neural explanations are given for psychophysical data concerning: contrast variations of dichoptic masking and the correspondence problem, the effect of interocular contrast differences on stereoacuity, Panum's limiting case, the Venetian blind illusion, stereopsis with polarity-reversed stereograms, da Vinci stereopsis, and various lightness illusions. By relating physiology to psychophysics, the model provides new functional insights and predictions about laminar cortical architecture.