Object perception is one of the most remarkable capacities of the primate brain. Owing to the large and indeterminate dimensionality of object space, the neural basis of object perception has been difficult to study and remains controversial. Recent work has provided a more precise picture of how 2D and 3D object structure is encoded in intermediate and higher-level visual cortices. Yet, other studies suggest that higher-level visual cortex represents categorical identity rather than structure. Furthermore, object responses are surprisingly adaptive to changes in environmental statistics, implying that learning through evolution, development, and also shorter-term experience during adulthood may optimize the object code. Future progress in reconciling these findings will depend on more effective sampling of the object domain and direct comparison of these competing hypotheses.