While so far we have considered physical systems of fermions, bosons or classical particles in thermodynamic equilibrium, or their response with respect to external perturbations (Chap.  8 ), it remains unclear how these systems reach the final equilibrium state in time and what are the characteristic timescales to achieve equilibrium. The answer to these questions requires the formulation of a consistent non-equilibrium dynamics that describes the explicit time evolution of the physical system. On the one hand, one can use the time-dependent Schrödinger equation and derive a system of kinetic equations using Green’s functions, on the other hand, the time evolution of the N-particle density matrix can also be considered directly in suitable approximations. In the following we will present the density-matrix formalism for weakly interacting N-particle fermion systems, that will provide the basis for the derivation of kinetic theories.