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In studying a number of processes, such as plasma confinement by magnetic field and magnetized-plasma confinement with walls, magnetic implosion of liners transiting to the plasma state during electric explosion, liner compression of magnetized plasma, magnetic-flux transfer through the insulator surface, etc., one needs to take into account surface discharges induced by magnetic-flux transfer toward the plasma or insulator (H-pressed discharge) or out of the insulator (H-thrown discharge). These discharges cause magnetic flux and energy losses (in the H-pressed discharge) or limit the energy fluxes delivered to the system and can lead to hydrogen plasma contamination with the insulator material (in the H-thrown discharge). The problem of magnetized-plasma cooling at the condensed matter/plasma interface is related to the H -pressed discharge problem. Such a problem as strong magnetic field diffusion into metal, which is very important for megagauss field generation and various applications, can also be attributed to the surface-discharge problems. If a metal is in contact with a dense magnetized plasma, surface discharges may occur near their interface, where the metal may explode and become a plasma. This paper presents the classification and the overview of the mentioned surface discharges in strong magnetic fields and considers general and specific features of these discharges.