How do we recognize objects despite changes in their appearance? The past three decades have been witness to intense debates regarding both whether objects are encoded invariantly with respect to viewing conditions and whether specialized, separable mechanisms are used for the recognition of different object categories. We argue that such dichotomous debates ask the wrong question. Much more important is the nature of object representations: What are features that enable invariance or differential processing between categories? Although the nature of object features is still an unanswered question, new methods for connecting data to models show significant potential for helping us to better understand neural codes for objects. Most prominently, new approaches to analyzing data from functional magnetic resonance imaging, including neural decoding and representational similarity analysis, and new computational models of vision, including convolutional neural networks, have enabled a much more nuanced understanding of visual representation. Convolutional neural networks are particularly intriguing as a tool for studying biological vision in that this class of artificial vision systems, based on biologically plausible deep neural networks, exhibits visual recognition capabilities that are approaching those of human observers. As these models improve in their recognition performance, it appears that they also become more effective in predicting and accounting for neural responses in the ventral cortex. Applying these and other deep models to empirical data shows great promise for enabling future progress in the study of visual recognition.