We present the definition and basic properties of the characteristic function of a random variable, which uniquely determines its distribution. We describe the relationship between the characteristic function and the moments of a random variable, as well as the relation between the convergence of a sequence of characteristic functions and the convergence in distribution of the corresponding sequence of variables. We also present and prove the Lévy-Cramér Theorem, which demonstrates that the method of characteristic functions is highly effective in proving various versions of the Central Limit Theorem. Finally, we define and describe the characteristic function of random vectors.