  The many-body physics at quantum phase transitions shows a subtle interplay
between quantum and thermal fluctuations, emerging in the low-temperature
limit. In this review, we first give a pedagogical introduction to the
equilibrium behavior of systems in that context, whose scaling framework is
essentially developed by exploiting the quantum-to-classical mapping and the
renormalization-group theory of critical phenomena at continuous phase
transitions. Then we specialize to protocols entailing the out-of-equilibrium
quantum dynamics, such as instantaneous quenches and slow passages across
quantum transitions. These are mostly discussed within dynamic scaling
frameworks, obtained by appropriately extending the equilibrium scaling laws.
We review phenomena at first-order quantum transitions as well, whose peculiar
scaling behaviors are characterized by an extreme sensitivity to the boundary
conditions, giving rise to exponentials or power laws for the same bulk system.
In the last part, we cover aspects related to the effects of dissipative
interactions with an environment, through suitable generalizations of the
dynamic scaling at quantum transitions. The presentation is limited to issues
related to, and controlled by, the quantum transition developed by closed
many-body systems, treating the dissipation as a perturbation of the critical
regimes, as for the temperature at the zero-temperature quantum transition. We
focus on the physical conditions giving rise to a nontrivial interplay between
critical modes and various dissipative mechanisms, generally realized when the
involved mechanism excites only the low-energy modes of the quantum
transitions.
