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Atmospheric-pressure gas discharge driven by high voltage pulses with fast rise-time and short duration has attracted significant attention for various plasma applications. In this paper, discharges were generated in a highly non-uniform electric field by point-plane gaps in open air by four repetitive nanosecond-pulse generators with repetition rate up to 1 kHz. The rise time of generators was 25 (generator #1), 15 (generator #2), 3 (generator #3), and 0.2 ns (generator #4) and a full width at half maximum was 40, 30-40, 5, and 1 ns, respectively. The experimental results show that there were typical discharge fashions, i.e., corona, diffuse, spark, or arc modes. The variables affecting the discharge characteristics, including the gap spacing and applied pulse parameters, were investigated. Especially, the diffuse discharges were investigated and discussed. With generator #1 at voltage 70-120 kV, characteristics of measured x-rays on the discharge modes were studied, and it indicates that counts of x-rays in a diffuse discharge are up to a peak value under the experimental conditions. With amplitude of voltage pulses in incident wave up to 18 (generator #3) and 12.5 kV (generator #4), runaway electron beam in low pressure helium, nitrogen, and air in a pulse-periodic mode of discharge with repetition rate up to 1 kHz was obtained. Electron beam was registered behind a thin foil in a pressure range from several to tens of Torr. X-ray radiation was obtained in a wide range of pressures, as well as at atmospheric pressure of helium, nitrogen, and air. Voltage pulses of positive and negative polarities were used. Generation of runaway electrons with pulses of positive polarity appeared because of reflected voltage pulses of reverse polarity.