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The characteristics of a corona discharge in atmospheric pressure air are studied using pulsed power generators that produce voltage pulses of different durations, polarities and shapes. The characteristics are measured in the single pulse, batch, and repetitively pulsed modes. It is shown that no matter what the voltage pulse polarity is, a corona discharge starts developing as a conical diffuse discharge near the electrode tip with a voltage rate of increase of ~1015 Vs-1 across an electrode of small curvature radius. With lower voltage rate of increase (~1013 Vs-1 or lower), one or several diffuse jets develop from this electrode. The diameter of the jets at their front is less than 1mm and depends on many factors (voltage pulse amplitude and increase, interelectrode gap width, pulse repetition rate, etc). It is found that at long voltage pulse durations, the radiation spectrum of the corona discharge changes, and the bands and lines of the material of the electrode appear in the UV region at 200-300 nm. It is demonstrated that a runaway electron beam in a corona discharge is generated and detected at a distance several times greater than the brightly glowing plasma region of the corona discharge. It is shown that x-rays are generated from a corona discharge at high pulse repetition rates of up to 1 kHz. [ABSTRACT FROM AUTHOR]