  We review one of the most versatile theoretical approaches to the study of
time-dependent correlated quantum transport in nano-systems: the
non-equilibrium Green's function (NEGF) formalism. Within this formalism, one
can treat, on the same footing, inter-particle interactions, external drives
and/or perturbations, and coupling to baths with a (piece-wise) continuum set
of degrees of freedom. After a historical overview on the theory of transport
in quantum systems, we present a modern introduction of the NEGF approach to
quantum transport. We discuss the inclusion of inter-particle interactions
using diagrammatic techniques, and the use of the so-called embedding and
inbedding techniques which take the bath couplings into account
non-perturbatively. In various limits, such as the non-interacting limit and
the steady-state limit, we then show how the NEGF formalism elegantly reduces
to well-known formulae in quantum transport as special cases. We then discuss
non-equilibrium transport in general, for both particle and energy currents.
Under the presence of a time-dependent drive - encompassing pump-probe
scenarios as well as driven quantum systems - we discuss the transient as well
as asymptotic behavior, and also how to use NEGF to infer information on the
out-of-equilibrium system. As illustrative examples, we consider model systems
general enough to pave the way to realistic systems. These examples encompass
one- and two-dimensional electronic systems, systems with electron-phonon
couplings, topological superconductors, and optically responsive molecular
junctions where electron-photon couplings are relevant.
