  We derive an exact equation of motion for the reduced density matrices of
individual subsystems of quantum many-body systems of any lattice dimension and
arbitrary system size. Our projection operator based theory yields a highly
efficient analytical and numerical approach. Besides its practical use it
provides a novel interpretation and systematic extension of mean-field
approaches and an adaption of open quantum systems theory to settings where a
dynamically evolving environment has to be taken into account. We show its high
accuracy for two significant classes of complex quantum many-body dynamics,
unitary evolutions of non-equilibrium states in closed and stationary states in
driven-dissipative systems.
