This is the fourth article in a series of surveys devoted to the scientific achievements of the Leningrad - Saint-Petersburg school of Probability and Statistics during the period from 1947 to 2017. It is devoted to the works on characterization of distributions and various limit theorems in Statistics including asymptotic efficiency of tests, criteria based on characterizations and normal approximation for L1 -norms of kernel density estimators. The characterization results are related to independence and equidistribution of linear and non-linear forms of the sample as well as to regression relations, admissibility and optimality of statistical estimators. When calculating the Bahadur efficiency, we give attention to the logarithmic asymptotics for large deviations of test statistics under the null hypothesis. We study also the conditions of local asymptotic optimality in Bahadur sense for various nonparametric tests.