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We review the basic issues of the initiation and development of the lightning discharge, which top the list of the most important and yet unsolved problems of atmospheric electricity. The main challenges of creating theoretical models are due to the fact that the value of the electric strength of atmospheric air exceeds the peak electric fields measured in thunderstorm clouds by approximately an order of magnitude. Among several concepts proposed to explain the process of initiation of a lightning discharge at different times, two ideas stand apart, namely, the hypothesis of the lightning birth caused by the initiation of a positive streamer from the surface of a hydrometeor and the hypothesis of lightning initiation due to the development of a runaway electron breakdown. However, none of these approaches has become universally recognized or dominant due to various difficulties. A fundamentally new mechanism of lightning discharge initiation, which has been proposed recently, is based on the noise-induced kinetic transition occurring in the stochastic field of charged hydrometeors. The proposed approach looks like a sequence of discharge activity transitions from small spatial scales to increasingly larger ones. One of the main features of the proposed hypothesis is that the generation of streamers is determined by the level of small- and mesoscale fluctuations of the electric field in the thunderstorm and is essentially independent of the large-scale field. In this case, the role of the large-scale field is to ensure interaction of the initiated streamers, when they start developing mainly in the direction specified by this field. In the final section of the review, we discuss the fundamental role of the polarity asymmetry in the processes of the initiation and further development of lightning discharges. [ABSTRACT FROM AUTHOR]