The lower bank of the superior temporal sulcus (TEs), part of the inferior temporal cortex, contains neurons selective for disparity-defined three-dimensional (3-D) shape. The large majority of these TEs neurons respond to the spatial variation of disparity, i.e., are higher-order disparity selective. To determine whether curved boundaries or curved surfaces by themselves are sufficient to elicit 3-D shape selectivity, we recorded the responses of single higher-order disparity-selective TEs neurons to concave and convex 3-D shapes in which the disparity varied either along the boundary of the shape, or only along its surface. For a majority of neurons, a 3-D boundary was sufficient for 3-D shape selectivity. At least as many neurons responded selectively to 3-D surfaces, and a number of neurons exhibited both surface and boundary selectivity. The second aim of this study was to determine whether TEs neurons can represent differences in second-order disparities along the horizontal axis. The results revealed that TEs neurons can also be selective for horizontal 3-D shapes and can code the direction of curvature (vertical or horizontal). Thus, TEs neurons represent both boundaries and surfaces curved in depth and can signal the direction of curvature along a surface. These results show that TEs neurons use not only boundary but also surface information to encode 3-D shape properties.