  Recent years have seen tremendous progress in the theoretical understanding
of quantum systems driven dissipatively by coupling them to different baths at
their edges. This was possible because of the concurrent advances in the models
used to represent these systems, the methods employed, and the analysis of the
emerging phenomenology. Here we aim to give a comprehensive review of these
three integrated research directions. We first provide an overarching view of
the models of boundary-driven open quantum systems, both in the weak and strong
coupling regimes. This is followed by a review of state-of-the-art analytical
and numerical methods, both exact, perturbative and approximate. Finally, we
discuss the transport properties of some paradigmatic one-dimensional chains,
with an emphasis on disordered and quasiperiodic systems, the emergence of
rectification and negative differential conductance, and the role of phase
transitions, and we give an outlook on further research options.
