  Experimental progress in atomic, molecular, and optical platforms in the last
decade has stimulated strong and broad interest in the quantum coherent
dynamics of many long-range interacting particles. The prominent collective
character of these systems enables novel non-equilibrium phenomena with no
counterpart in conventional quantum systems with local interactions. Much of
the theory work in this area either focussed on the impact of variable-range
interaction tails on the physics of local interactions or relied on
mean-field-like descriptions based on the opposite limit of all-to-all
infinite-range interactions. In this Report, we present a systematic and
organic review of recent advances in the field. Working with prototypical
interacting quantum spin lattices without disorder, our presentation hinges
upon a versatile theoretical formalism that interpolates between the few-body
mean-field physics and the many-body physics of quasi-local interactions. Such
a formalism allows us to connect these two regimes, providing both a formal
quantitative tool and basic physical intuition. We leverage this unifying
framework to review several findings of the last decade, including the peculiar
non-ballistic spreading of quantum correlations, counter-intuitive slowdown of
entanglement dynamics, suppression of thermalization and equilibration,
anomalous scaling of defects upon traversing criticality, dynamical phase
transitions, and genuinely non-equilibrium phases stabilized by periodic
driving. The style of this Report is on the pedagogical side, which makes it
accessible to readers without previous experience in the subject matter.
