  Recent studies of interacting systems of quantum spins, ultracold atoms and
correlated fermions have shed a new light on how isolated many-body systems can
avoid rapid equilibration to their thermal state. It has been shown that many
such systems can "weakly" break ergodicity: they possess a small number of
non-thermalising eigenstates and/or display slow relaxation from certain
initial conditions, while majority of other initial states equilibrate fast,
like in conventional thermalising systems. In this chapter, we provide a
pedagogical introduction to weak ergodicity breaking phenomena, including
Hilbert space fragmentation and quantum many-body scars. Central to these
developments have been the tools based on quantum entanglement, in particular
matrix product states and tangent space techniques, which have allowed to
analytically construct non-thermal eigenstates in various non-integrable
quantum models, and to explore semiclassical quantisation of such systems in
the absence of a large-N or mean field limit. We also discuss recent
experimental realisations of weak ergodicity breaking phenomena in systems of
Rydberg atoms and tilted optical lattices.
