The theory of algebraic invariants has found insufficient attention in Russian mathematical literature. The book by Alekseev [I I]*, written in 1899, is largely out of date, while individual chapters in certain textbooks on algebra (Sushkevich, Bocher, etc.) written later, give only the beginnings of the theory. The present book is to fill this gap. Its essential special feature is wide utilization of classical methods as well as of the basic concepts and notation of tensor algebra; this makes it possible to present all problems at once in as general a form as possible. In addition, the Author believes that only by this means can one succeed in bringing full clarity to the problem of Aronhold's symbolic (?18). Throughout I have attempted not to leave out of sight the close link which exists between invariant theory and geometry; the extensive geometric introduction of Chapter 1 also serves this purpose. Chapter 11 which is concerned with the foundations of tensor algebra also bears an introductory character. The general propositions of the theory of algebraic invariants are given in Chapters III and IV, the most important particular results in Chapters V-VII; among these Chapter V is more classical in spirit. At the end of each section, I have given exercises of which there are altogether approximately 500. They serve a double purpose in that some explain the preceding work, others consider problems treated insufficiently in the main text. At times, a set of problems will present some substantial section of theory, for example, the exercises following ¨ 3, 10-14, 25 contain the classification of binary forms of fourth order in the real domain. Certain problems relate to the subsequent text; in that case they have been provided with asterisks. In collecting the exercises, I have made use of the references [3, 11, 13, 14, 16, 19] as well as of H. Beck's ,Koordinaten Geometric". However, a substantial number of the problems have been published here for the first time. In conclusion, I wish to express my gratitude to Ia. S. Dubnov, who has studied the manuscript of Chapter I and made a number of valuable comments, and to M. G. Freidinii for his painstaking editorial work. G. B. Gurevich.