The hollow cathode discharge is a kind of plasma formation scheme in which plasma is formed inside a hollow structure, the cathode, with current to a nearby anode of arbitrary shape. In this scheme, electrons reflex radially within the hollow cathode, establishing an efficient ionization mechanism for gas within the cavity. An existence condition for the hollow cathode effect is that the electron mean-free-path for ionization is of the order of the cavity radius. Thus the size of this kind of plasma source must decrease as the gas pressure is increased. In fact, the hollow cathode effect can occur even at atmospheric pressure for cathode diameters of order 10-100 $mu extrm{m}$. That is, the "natural" operating pressure regime for a "micro hollow cathode discharge" is atmospheric pressure. This kind of plasma source has been the subject of increasing research activity in recent years. A number of geometric variants have been explored, and operational requirements and typical plasma parameters have been determined. Large arrays of individual tiny sources can be used to form large-area, atmospheric-pressure plasma sources. The simplicity of the method and the capability of operation without the need for the usual vacuum system and its associated limitations, provide a highly attractive option for new approaches to many different kinds of plasma applications, including plasma surface modification technologies. Here we review the background work that has been carried out in this new research field.