This paper provides a systematic study of order statistics drawn from discrete parent distributions. New procedures are followed for the derivation of the distribution of the r^th order statistic x_(r) and of the joint distribution of x_(r), x_(s) (s > r), that is, we first derive the cumulative probability distribution, from which the probability distribution comes directly. This approach is easier than the usual method, moreover the formulae for the c.d.f. derived in this way can be easily computed. To get the moments of X_(r), we use convenient formulae involving the tails of the c.d.f. of X_(r) rather than the p.d.f. of X_(r). The moments are then readily derived from general results for discrete distributions. We show the analogy between the results in the continuous and discrete cases. Applications to three discrete distributions are given. We consider some results on uncorrelated statistics which were established in the continuous case and show that the same results hold also for the discrete case. Many recurrence relations between moments of order statistics are derived in the discrete case yielding the same results as previously given by Govindarajulu (1963) and Sillitto (1951, 1964) in the continuous case. / M.S.