We introduce new techniques to extend superrigidity theory beyond the scope of Lie or algebraic groups. We construct a cohomological invariant which accounts for, and generalizes, all known superrigidity results for actions on negatively curved spaces. Together with a new vanishing result and the machinery of bounded cohomology, this enables us to prove a general superrigidity theorem for actions of irreducible lattices on spaces of negative curvature. We also prove a cocycle version à la Zimmer