A new class of stationary solutions in the theory of glow discharges and plasma-cathode interaction in arc discharges has been found in the course of the past 15 years. These solutions exist simultaneously with the solution given in textbooks, which describes a discharge mode with a uniform or smooth distribution of current over the cathode surface, and describe modes with various configurations of cathode spots: normal spots on glow cathodes, patterns of multiple spots recently observed on cathodes of glow microdischarges, spots on arc cathodes. In particular, these solutions show that cathode spots represent a manifestation of self-organization caused by basic mechanisms of near-cathode space-charge sheath; another illustration of the richness of the gas discharge science. As far as arc cathodes are concerned, the new solutions have proved relevant for industrial applications. This talk is dedicated to reviewing the multiple solutions obtained to date and the physics of plasma-cathode interaction in arc discharges in ambient gas and vacuum and in glow discharges revealed by these solutions.