This is the first 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 traditional for St. Petersburg topic of limit theorems for sums of independent random variables. We discuss classical limit theorems: the law of large numbers, the central limit theorem, and the law of the iterated logarithm, as well as the circle of important related problems that emerged in the second half of the twentieth century. The latter include approximation for the distributions of sums of independent summands by infinitely divisible distributions, an estimate of the accuracy of strong Gaussian approximation for such sums, and the weak almost sure convergence empirical measures generated by a sequence of sums of independent random variables and vectors.