The theory of finite-dimensional algebras is one of the most fundamental domains of modern algebra, applied in several other parts of mathematics and in theoretical physics. This book, written by two of the leading researchers in the field and revised and augmented for the English edition, was translated from the Russian by a third leading specialist, who has contributed to it an appendix. The book presents both the basic classical theory and more recent results closely related to current research (some category theory including Morita's theorem, schemes of quivers and tensor algebras, duality, quasi-Frobenius, hereditary, serial algebras). The only prior knowledge assumed of the reader is linear algebra and, in places, a little group theory. Each chapter includes a series of exercises, illustrating the content and introducing more refined results: for the more complicated ones, hints for the solution are given - thus the book can be used as a textbook in class or for self-study, and as an up-to-date reference to the field.