A theory of the hollow cathode arc is given which explains the physical mechanisms of operation of these discharges. The following basic phenomena are studied: the gas flow inside the cathode; the formation of the internal plasma column by ionization of the neutral gas by the primary electrons emitted by the cathode, which are accelerated in the cathode sheath; the transport of the ions towards the cathode wall; the transport of current and of kinetic energy towards the exit of the cathode, by the Maxwellian electrons of the plasma; and, finally, the thermal equilibrium of the cathode wall. This theory gives the axial variation of the gas pressure and wall temperature, the electron temperature, the mean number of excitations and of ionizations produced per primary electron, and the radial and the axial variation of the plasma density and plasma potential. The field‐enhanced thermionic effect constitutes the basic emission mechanism of the cathode, the wall being heated mainly by ion bombardment. Theoretical results for discharges in argon with a tantalum cathode are in excellent agreement with all the available experimental results.