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The book consists of 15 chapters written by experts in the fields who are engaged in the theoretical and mathematical simulations of the physical process associated with lightning and other electrical discharges. Starting with the basics of electromagnetic theory, the reader is first introduced to the subject of measurements of electromagnetic fields from lightning. After this the reader is introduced to the modelling of charge generation in thunderclouds, the generation of X-ray and gamma bursts by runaway electrons in the thundercloud fields and the initiation of lightning flashes. One chapter of the book illustrates how to model stepped leaders, connecting leaders and the final attachment. This chapter takes into account also the simulation of upward initiated lightning flashes from tall structures and space vehicles under the influence of the thundercloud electric fields. Several chapters discuss the various methods available to model lightning return strokes including, antenna models, transmission line models and various engineering models. They also provide mathematical and simulation tools necessary (a) to calculate the electromagnetic fields from lightning flashes, (b) to study the propagation of lightning electromagnetic fields over grounds of different conductivities and also around the Earth (c) to simulate Schumann resonances and (d) to study the interaction of lightning flashes with power lines and tall structures. The book also provides the reader with information to simulate generation of NOx from lightning flashes.