Abstarct: Probability and moment inequalities plays a key role in studying limiting behavior of partial sums of random variables. When the value of probability of tail or moment of sum of random variables is not calculable, various inequalities are presented. In most of them, the independence condition may not hold, then, the generalization of probability and moment inequalities must be used to verify limiting properties of sum of random variables. Specially, acceptable variables, negatively superadditive dependent, j-dependent as an applications may be used.