This chapter is devoted in a systematic study of a weak convergence of sequences of random variables. It is shown the equivalence between a weak convergence and convergence in distribution. It is shown that a weak compactness and tightness for families of probability distributions (Prokhorov’s theorem) is given. It is discussed a connection between characteristic functions and distributions of random variables. The method of characteristic functions is applied to prove the Central limit theorem (CLT) central limit theorem (CLT) for sums of independent identically distributed random variables (see Bulinski and Shiryayev, Theory of stochastic processes, 2005; Jacod and Protter, Probability essentials, 2003; Krylov, Introduction to the theory of random processes, 2002, and Shiryaev, Probability, 1996).