Undercooling and crystallization kinetics are recognized increasingly as important processes controlling the final textures and compositions of minerals as well as the physicochemical state of magmas during ascent and emplacement. Within a single volcanic unit, phenocrysts, microphenocrysts and microlites can span a wide range of compositions, develop complex zoning patterns, and show intricate textures testifying to crystallization far from equilibrium. These petrographic complexities are not associated necessarily with magma chamber processes such as mixing or mingling of distinctly different bulk compositions but, rather, may be caused by variable degrees of initial magma-undercooling and the evolution of undercooling through time. Heat-dissipation and decompression are the most effective driving forces of cooling and volatile loss that, in turn, exert a primary control on the solidification path of magma. Understanding these kinetic aspects over the temporal and spatial scales at which volcanic processes occur is therefore essential to interpret correctly the time-varying environmental conditions recorded in igneous minerals.This contribution aims to summarize and integrate experimental studies pertaining to the crystallization of magmas along kinetic or time-dependent pathways, where solidification is driven by changes in temperature, pressure and volatile concentration. Fundamental concepts examined in the last decades include the effect of undercooling on crystal nucleation and growth as well as on the transition between interface-and diffusion-controlled crystal growth and mass transfer occurring after crystals stop growing. We summarize recent static and dynamic decompression and cooling experiments that explore the role of undercooling in syn-eruptive crystallization occurring as magmas ascend in volcanic conduits and are emplaced at the surface. The ultimate aim of such studies is to decode the textural and compositional information within crystalline phases to place quantitative constraints on the crustal transport, ascent and emplacement histories of erupted and intrusive magmas.Magma crystallization under dynamic conditions will be assessed also through a comparative description of the disequilibrium features in minerals found in experimental and natural materials. A variety of departures from polyhedral growth, including morphologies indicating crystal surface instability, dendritic structures, sector zoning and growth twins are linked to the rate at which crystals grow. These have implications for the entrapment of melt inclusions and plausibility for interpreting the growth chronology of individual crystals. A simple ''tree-ring'' model, in which the oldest part of the crystal lies at the centre and the youngest at the rim, is not an appropriate description when growth is non-concentric. Further, deviation from chemical