  A density matrix {\rho}(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 {\Xi}(t) is composed of two objects, {\rho}(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 projective
and weak value measurements 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
post-selected quantum states [Y. Aharonov and L. Vaidman, J. Phys. A: Math.
Gen. 24, 2315 (1991)] to systems subject to any quantum measurement scenario,
any coherent evolution, and any Markovian dissipation processes.
