Although object recognition is fundamental to our behavior and seemingly effortless, it is a remarkably challenging computational problem because the visual system must somehow tolerate tremendous image variation produced by different views of each object (the "invariance" problem). To understand how the primate brain accomplishes this remarkable feat, we must understand how sensory input is transformed from an initial neuronal population representation (a photograph on the retina), to a new, remarkably powerful form of neuronal population representation at the highest level of the primate ventral visual stream (inferior temporal cortex, IT). In this talk, I will review our results on the ability of the IT population representation to support position-, scale- and clutter-tolerant recognition. I will present a geometric perspective for thinking about how this ventral visual stream constructs this representation ("untangling" object manifolds). Finally, I will show our recent neurophysiologic and psychophysical results that suggest that this untangling is driven by the spatiotemporal statistics of unsupervised natural visual experience. Our long term goal is to use the understanding of this biological computation to inspire artificial vision systems, to aid the development of visual prosthetics, to provide guidance to molecular approaches to repair lost brain function, and to obtain deep insight into how the brain represents sensory information in a way that is highly suited for cognition and action.