A density matrix ρ(t) yields probabilistic information about the outcome of measurements on a quantum system. We introduce here the past quantum state, which, at time T, accounts for the state of a quantum system at earlier times t<T. The past quantum state Ξ(t) is composed of two objects, ρ(t) and E(t), conditioned on the dynamics and the probing of the system until t and in the time interval t, T, respectively. The past quantum state is characterized by its ability to make better predictions for the unknown outcome of any measurement at t than the conventional quantum state at that time. On the one hand, our formalism shows how smoothing procedures for estimation of past classical signals by a quantum probe M. Tsang, Phys. Rev. Lett. 102, 250403 (2009) apply also to describe the past state of the quantum system itself. On the other hand, it generalizes theories of pre- and postselected quantum states Y. Aharonov and L. Vaidman, J. Phys. A 24, 2315 (1991) to systems subject to any quantum measurement scenario, any coherent evolution, and any Markovian dissipation processes.