In applied statistics, one observes a random quantity X, a number of times and based on these observations, one would like to conclude facts about the distribution function F(x) of X. The only method of finding distribution function F(x) exactly, which avoids the subjective choice, is a characterization theorem. Important consequence of characterization theorem is that these results help us in better understanding the structures and implications of the choice of distribution for a special problem. Regarding moments of generalized order statistics (gos), a concept introduced by Kamps (1995). Order and record statistics are particular cases of gos. Recurrence relations have great importance due to these reasons: (i). Reduce the amount of direct computations and hence reduce the time and labour (ii). They express the higher order moments in terms of the lower order moments and hence make the evaluation of higher order moments easy (iii). Provide simple checks to test the accuracy of computation of moments of order statistics. This book provides some new characterization and moments of order statistics, records and gos.