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The book presents a review of the work done on the basic distribution theory of discrete order statistics, recurrence relations and identities for the order statistics in the i.i.d case and discuss some specific discrete distributions in detail. The introductory chapter gives a brief and comprehensive account of the historical development of the discrete order statistics. Second chapter deals with the distributions of order statistics and sample spacing, and dependence structure of discrete order statistics. It also deals with the distributions of order statistics of a sample drawn at random without replacement. In the third chapter, several recurrence relations and identities for the single and product moments of order statistics in a sample of size n from an arbitrary discrete distribution are presented. Making use of these recurrence relations, it is shown that it is sufficient to evaluate just two single moments and (n -1)/2 product moments when n is odd and two single moments and (n -2)/2 product moments when n is even, in order to evaluate the first, second and product moments of order statistics in a sample of size n, given these moments in samples of size n-1 and less.