Grain rotation is commonly observed during the evolution of microstructures in polycrystalline materials of different kinds, including metals, ceramics, and colloidal crystals. It is widely accepted that interface migration in these systems is mediated by the motion of line defects with step and dislocation character, i.e., disconnections. We propose a crystallography-respecting continuum model for arbitrarily curved grain boundaries or heterophase interfaces, accounting for the disconnections' role in grain rotation. Numerical simulations demonstrate that changes in grain orientations, as well as interface morphology and internal stress field, are associated with disconnection flow. Our predictions agree with molecular dynamics simulation results for pure capillarity-driven evolution of grain boundaries and are interpreted through an extended Cahn-Taylor model.