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A fundamental and largely unanswered question in neuroscience is whether extrinsic connectivity and function are closely related at a fine spatial grain across the human brain. Using a novel approach, we found that the anatomical connectivity of individual gray-matter voxels (determined via diffusion-weighted imaging) alone can predict functional magnetic resonance imaging (fMRI) responses to 4 visual categories (faces, objects, scenes, and bodies) in individual subjects, thus accounting for both functional differentiation across the cortex and individual variation therein. Furthermore, this approach identified the particular anatomical links between voxels that most strongly predict, and therefore plausibly define, the neural networks underlying specific functions. These results provide the strongest evidence to date for a precise and fine-grained relationship between connectivity and function in the human brain, raise the possibility that early-developing connectivity patterns may determine later functional organization, and offer a method for predicting fine-grained functional organization in populations who cannot be functionally scanned.