A number of general information from mathematical analysis and the theory of probability distributions are given, as well as the necessary algorithms for computer generation of one-dimensional discrete random variables. Important general classes of one-dimensional discrete quantities are introduced, including families of mixed and composite random variables. Properties of binomial, Poisson, negative binomial, geometric, hypergeometric, logarithmic distributions families are considered in detail. Several dozen related families of discrete random variable distributions are considered in less detail.