Information that is shared across brains is encoded in idiosyncratic fine-scale functional topographies. Hyperalignment jointly models shared information and idiosyncratic topographies. Pattern vectors for neural responses and connectivities are projected into a common, high-dimensional information space, rather than being aligned in a canonical anatomical space. Hyperalignment calculates individual transformation matrices that preserve the geometry of pairwise dissimilarities between pattern vectors. Individual cortical topographies are modeled as mixtures of overlapping, individual-specific topographic basis functions, rather than as contiguous functional areas. The fundamental property of brain function that is preserved across brains is information content, rather than the functional properties of local features that support that content.