Despite tremendous variation in the appearance of visual objects, primates can recognize a multitude of objects, each in a fraction of a second, with no apparent effort. However, the brain mechanisms that enable this fundamental ability are not understood. Drawing on ideas from neurophysiology and computation, we present a graphical perspective on the key computational challenges of object recognition, and argue that the format of neuronal population representation and a property that we term 'object tangling' are central. We use this perspective to show that the primate ventral visual processing stream achieves a particularly effective solution in which single-neuron invariance is not the goal. Finally, we speculate on the key neuronal mechanisms that could enable this solution, which, if understood, would have far-reaching implications for cognitive neuroscience.