*There will be snacks and drinks at 4pm and the talk starts at 4:10pm*
We show that time-reflection symmetry in periodically driven (Floquet) quantum systems enables an inherently nonequilibrium phenomenon structurally similar to quantum-mechanical sypersymmetry. In particular, we find Floquet analogues of the Witten index that place lower bounds on the degeneracies of states with quasienergies 0 and π. We provide a simple class of disordered, interacting, and ergodic Floquet models with an exponentially large number of states at quasienergies 0,π, which are robust as long as the time-reflection symmetry is preserved. Floquet supersymmetry manifests itself in the evolution of certain local observables as a period-doubling effect with dramatic finite-size scaling, providing a clear signature for experiments.