The characterization of distribution is useful for selection of adequate distribution to describe the observed values obtained in an experiment and is one of the methods of finding the distribution. Chapter 3 and 4 are concerned with the characterization developed by Kemp and Kemp (2004) and Ahmad and Roohi (2004). In Chapter 5, the recurrence relations between ordinary moments are established. A general characterization theorem, based on recurrence relation of ordinary moments is derived for a general class of discrete distributions. Chapter 6 deals with the recursive relations of factorial moments obtained by successive differentiation of factorial moment generating functions. In Chapters 7, 8, and 9 the theorems are then applied to numerous discrete probability distributions to provide specific characterizations for each one of them. Since information concerning moments is more often available than the knowledge of probability distribution as a whole, we expect these properties to be useful in dealing with the practical problems.