THE object of this tract is to enable mathe-X maticians who are not specialists in the theory of numbers to learn some of its non-elementary results and methods without too great an effort. It assumes no previous knowledge of the subject beyond what is given in Hardy and Wright's "An Introduction to the Theory of Numbers". The tract consists of three chapters. The first deals with Riemann's zeta function and a refinement of the asymptotic formula for the number of primes not exceeding a given integer. The second chapter deals with primes in arithmetical progression, using Dirichlet's L functions, with many theorems on characters. The third chapter deals with the representation of an odd number as a sum of three primes, using the methods of Hardy and Littlewood, and an extension of them by Vinogradoff. The tract concludes with a list of ninety-six theorems and formulae for reference, and with a very short index. The author seems to have done his best to simplify a difficult subject, but those new to it may be surprised at the complexity of the methods necessary to establish a simple result.