The many-body physics at quantum phase transitions shows a subtle interplaybetween quantum and thermal fluctuations, emerging in the low-temperaturelimit. In this review, we first give a pedagogical introduction to theequilibrium behavior of systems in that context, whose scaling framework isessentially developed by exploiting the quantum-to-classical mapping and therenormalization-group theory of critical phenomena at continuous phasetransitions. Then we specialize to protocols entailing the out-of-equilibriumquantum dynamics, such as instantaneous quenches and slow passages acrossquantum transitions. These are mostly discussed within dynamic scalingframeworks, obtained by appropriately extending the equilibrium scaling laws.We review phenomena at first-order quantum transitions as well, whose peculiarscaling behaviors are characterized by an extreme sensitivity to the boundaryconditions, giving rise to exponentials or power laws for the same bulk system.In the last part, we cover aspects related to the effects of dissipativeinteractions with an environment, through suitable generalizations of thedynamic scaling at quantum transitions. The presentation is limited to issuesrelated to, and controlled by, the quantum transition developed by closedmany-body systems, treating the dissipation as a perturbation of the criticalregimes, as for the temperature at the zero-temperature quantum transition. Wefocus on the physical conditions giving rise to a nontrivial interplay betweencritical modes and various dissipative mechanisms, generally realized when theinvolved mechanism excites only the low-energy modes of the quantumtransitions.