This book deals with the asymptotic behaviour of probabilities of rare events related to large deviations of random walks whose jump distributions do not vanish too fast at infinity and are regularly behaved. The monograph presents an up-to-date, unified and systematic exposition of the field. Most of the results presented are appearing in a monograph for the first time and a good proportion of them were obtained by the authors. The key concepts of the monograph have a crucial standing in modern probability: the Random walk is a classical, probabilistic object of great importance in mathematical statistics, risk and queuing theory. Large deviation and rare events are of great interest in many applied areas, since computing the large deviation probabilities enables one to establish, for example, small error probabilities in statistics or ruin probabilities in risk theory. Regular distributions offer an important alternative to the classic case of distributions decaying exponentially fast at infinity, which unfortunately fails in many applied problems. The book presents some beautiful and useful mathematics that may attract a number of probabilists to the large deviations topic in probability.