  We review selected advances in the theoretical understanding of complex
quantum many-body systems with regard to emergent notions of quantum
statistical mechanics. We cover topics such as equilibration and thermalisation
in pure state statistical mechanics, the eigenstate thermalisation hypothesis,
the equivalence of ensembles, non-equilibration dynamics following global and
local quenches as well as ramps. We also address initial state independence,
absence of thermalisation, and many-body localisation. We elucidate the role
played by key concepts for these phenomena, such as Lieb-Robinson bounds,
entanglement growth, typicality arguments, quantum maximum entropy principles
and the generalised Gibbs ensembles, and quantum (non-)integrability. We put
emphasis on rigorous approaches and present the most important results in a
unified language.
