  Over the past decades, a great body of theoretical and mathematical work has
been devoted to random-matrix descriptions of open quantum systems. In these
notes, based on lectures delivered at the Les Houches Summer School "Stochastic
Processes and Random Matrices" in July 2015, we review the physical origins and
mathematical structures of the underlying models, and collect key predictions
which give insight into the typical system behaviour. In particular, we aim to
give an idea how the different features are interlinked. The notes mainly focus
on elastic scattering but also include a short detour to interacting systems,
which we motivate by the overarching question of ergodicity. The first chapters
introduce general notions from random matrix theory, such as the ten
universality classes and ensembles of hermitian, unitary, positive-definite and
non-hermitian matrices. We then review microscopic scattering models that form
the basis for statistical descriptions, and consider signatures of random
scattering in decay, dynamics and transport. The last chapter briefly touches
on Anderson localization and localization in interacting systems.
