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This paper contains a review of modern physical models of vacuum arc cathode spots in the discussion of which the main attention has been devoted to the question of the cathode spot current density j. Experimental features of current density measurements and possible errors with the use of different techniques are discussed in detail. The physical limitations due to transfer of high current densities through the near-cathode plasma have been considered and the criteria which limit the highest and lowest possible values of current density are suggested. In addition, the possibility of obtaining high current density emission in the " metal-dielectric-vacuum" system was examined.