In the area of quantum statistics, I show that a rigorous and systematic mathematical basis of the computational formulas of statistical physics does not require a special unwieldy analytical apparatus (the method of Darwin-Fowler), but may be obtained from an elementary application of the well-developed limit theorems of the theory of probability. Apart from its purely scientific value, which is evident and requires no comment, the possibility of such an appli­ cation is particularly satisfying to Soviet scientists, since the study of these limit theorems was founded by P. L. Chebyshev and was developed fur­ ther by other Russian and Soviet mathematicians. The fact that these theorems can form the analytical basis for all the computational formulas of statistical physics once again demonstrates their value for applications.This monograph, like my first book, is devoted entirely to the mathe­ matical method of the theory and is in no way a complete physical treatise. In fact, no concrete physical problem is considered. The book is directed primarily towards the mathematical reader. However, I hope that the physicist who is concerned with the mathematical apparatus of his science will find something in it to interest him.