Since the 1930s, the analytic theory of numbers has been transformed by the influence of I. M. Vinogradov. The Method of Trigonometrical Sums in the Theory of Numbers, which led to remarkable new results, testifies to its author's supreme ingenuity and to the effectiveness of his methods. Starting with a discussion of general lemmas, the text advances to an investigation of Waring's problem, including explorations of the singular series, the contribution of the basic intervals, and an estimate for G(n). Further topics include approximation by the fractional parts of the values of a polynomial, estimates for Weyl sums, the asymptotic formula in Waring's problem, the distribution of the fractional parts of the values of a polynomial, estimates for the simplest trigonometrical sums with primes, and Goldbach's problem. Recent research on the type of problems studied by Vinogradov indicates that his methods remain among the most powerful. This text will prove of enormous benefit to upper-level undergraduates and graduate students.