  This article presents an overview on recent progress in the theory of
nonequilibrium Green functions (NEGF). NEGF, presently, are the only
\textit{ab-initio} quantum approach that is able to study the dynamics of
correlations for long times in two and three dimensions. However, until
recently, NEGF simulations have mostly been performed with rather simple
selfenergy approximations such as the second-order Born approximation (SOA).
While they correctly capture the qualitative trends of the relaxation towards
equilibrium, the reliability and accuracy of these NEGF simulations has
remained open, for a long time. Here we report on recent tests of NEGF
simulations for finite lattice systems against exact-diagonalization and
density-matrix-renormalization-group benchmark data. The results confirm the
high accuracy and predictive capability of NEGF simulations---provided
selfenergies are used that go beyond the SOA and adequately include strong
correlation and dynamical-screening effects. We present a selfcontained
introduction to the theory of NEGF and give an overview on recent numerical
applications to compute the ultrafast relaxation dynamics of correlated
fermions. In the second part we give a detailed introduction to selfenergies
beyond the SOA. Important examples are the third-order approximation, the
\GWAx, the \TMA and the fluctuating-exchange approximation. We give a
comprehensive summary of the explicit selfenergy expressions for a variety of
systems of practical relevance, starting from the most general expressions and
the Feynman diagrams, and including also the important cases of diagonal basis
sets, the Hubbard model and the differences occuring for bosons and fermions.
With these details, and information on the computational effort and scaling
with the basis size and propagation duration, an easy use of these
approximations in numerical applications is made possible.
