We establish the existence of a topological classification of many-particle quantum systems undergoing unitary time evolution. The classification naturally inherits phenomenology familiar from equilibrium -- it is robust against disorder and interactions, and exhibits a non-equilibrium bulk-boundary correspondence, which connects bulk topological properties to the entanglement spectrum. We explicitly construct a non-equilibrium classification of non-interacting fermionic systems with

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... symmetries in all spatial dimensions (the 'ten-fold way'), which differs from its equilibrium counterpart. Direct physical consequences of our classification are discussed, including important ramifications for the use of topological zero-energy bound states in quantum information technologies.