This review addresses the kinetics of spontaneous boiling-up of liquefied gases and their solutions. It discusses the theories of thermal and quantum nucleation in metastable liquids. The experimental methods for studying the nucleation kinetics are outlined. The experimental data on the attainable superheating temperature for cryogenic liquids and solutions of liquefied gases are presented in a wide range of pressures, including negative ones. The properties of new-phase nuclei near the boundary of attainable superheating are discussed. The kinetics of initiated and heterogeneous nucleation is considered. The experimental data on detection of quantum tunneling of nuclei are presented. The experimental data are compared with the theories of thermal and quantum nucleation.

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