Abstract: A review on distributions of runs and succession events under various situations is given. Recently, a considerable number of studies have been made in this field with applications to reliability theory of consecutive systems, start--up demonstration tests, etc. In this paper, exact distribution theory is mainly surveyed such as discrete distributions of order k. The classical discrete distributions such as binomial, negative binomial and Poisson distributions have been modified to become suitable statistical models for practical and complicated applications. There are various modifications on underlying seqnences of trials or sampling schemes. Modifications on ways of connting the number of runs have been also considered to adjust the statistical model to practical situations.