This is a translation of the Russian book ПРЕДЕЛЬНЫЕ РАСПРЕДЕЛЕНИЯ ДЛЯ СУММ НЕЗАВИСИМЫХ СЛУЧАЙНЫХ ВЕЛИЧИН (1949). There are various points of contact with the treatises by P. Levy [76] and by H. Cramer [21], but much of the material in the book has been hitherto available only in periodical articles, many of which are in Russian. The systematic account presented here combines generality with simplicity, making some of the most important and difficult parts of the theory of probability easily accessible to the reader. Beyond a knowledge of the calculus on the level of, say, Hardy's Pure Mathematics, the book is formally self-contained. However, a certain amount of mathematical maturity, perhaps a touch of single-minded perfectionism, is needed to penetrate the depth and appreciate the classic beauty of this definitive work.