Theoretical and experimental studies of various modes of corona discharge operation in atmospheric pressure air are presented in this short review. The original results of modeling negative corona discharges are presented, taking into account the non-stationary plasma-chemical kinetics of charged particles in air plasma. The space–time evolution of the discharge in needle-to-plane geometry is investigated and analyzed. Several stages of discharge development are revealed from the moment of initiation of a low-negative current corona to the quasi-stationary mode of a glow discharge. Experimental data of the authors are presented. Modern technology and diagnostic equipment with a wide variation of the main parameters (the shape and polarity of the applied voltage, the type of gap, etc.) was used. The measurement of the optical characteristics of the plasma glow was carried out with high spatial resolution. Corona discharge current pulse profiles in the air at atmospheric pressure have been recorded with subnanosecond time resolution. With a positive polarity of the pin electrode and high voltage, a transition from a spherical streamer initiating a corona discharge to a cylindrical streamer is shown. The author’s results are rigorously evaluated through a critical comparison with findings from other research groups.