] and Mitsuo Shoji [a] With recent progress in theoretical and computational developments, real-time quantum many-body dynamics has become feasible. We first review the quantum dynamics researches including semiclassical treatments by quantum chemists. As one of the latest topics, we introduce the quantal cumulant dynamics (QCD) method, which solves hierarchical equations of motion for cumulant variables together with classical coordinates and momenta, as a natural extension of classical mechanics. We then demonstrate the applications of realtime QCD for molecular vibrations, dynamic isotope effects on proton transfer reactions, and quantum effects on dynamic structures of a quantum Morse cluster. V C 2014 Wiley Periodicals, Inc.