The problem of moments is of fairly old origin, but it received its first systematic treatment in the works of Tchebycheff, Markoff, Stieltjes, and, later, Hamburger, Nevanlinna, M. Riesz, Hausdorff, Carleman, and Stone. The subject has an extensive literature, but has not been treated in book or monograph form. In view of the considerable mathematical (and also practical) interest of the moment problem it appeared to the authors desirable to submit such a treatment to a wide mathematical public. In the present monograph the main attention is given to the classical moment problem, and, with the exception of a few remarks concerning the trigonometrical moment problem, no mention is made of various generalizations and modifications, important as they may be. Furthermore, lack of space did not permit the treatment of important developments of Carleman and Stone based on the theory of singular integral equations and operators in Hilbert space. On the other hand, a special chapter is devoted to the theory of approximate (mechanical) quadratures, which is intimately related to the theory of moments and in many instances throws additional light on the situation.