  The study of nonequilibrium phenomena in correlated lattice systems has
developed into an active and exciting branch of condensed matter physics. This
research field provides rich new insights that could not be obtained from the
study of equilibrium situations, and the theoretical understanding of the
physics often requires the development of new concepts and methods. On the
experimental side, ultra-fast pump-probe spectroscopies enable studies of
excitation and relaxation phenomena in correlated electron systems, while
ultra-cold atoms in optical lattices provide a new way to control and measure
the time-evolution of interacting lattice systems with a vastly different
characteristic timescale compared to electron systems. A theoretical
description of these phenomena is challenging because, firstly, we have to
compute the quantum-mechanical time-evolution of many-body systems out of
equilibrium, and secondly, deal with strong-correlation effects which can be of
nonperturbative nature. In this review, we discuss the nonequilibrium extension
of the dynamical mean field theory (DMFT), which treats quantum fluctuations in
the time domain and works directly in the thermodynamic limit. The method
reduces the complexity of the calculation via a mapping to a self-consistent
impurity problem. Particular emphasis is placed on a detailed derivation of the
formalism, and on a discussion of numerical techniques, which enable solutions
of the effective nonequilibrium DMFT impurity problem. We summarize the
insights gained into the properties of the infinite-dimensional Hubbard model
under strong non-equilibrium conditions. These examples illustrate the current
ability of the theoretical framework to reproduce and understand fundamental
nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators,
photo-doping, and collapse-and-revival oscillations in quenched systems.
