From the reviews of the first printing, published as Volume 73 of the Encyclopaedia of Mathematical Sciences:"... During the past twenty years, the representation theory of finite dimensional algebras has developed rapidly. The book presented serves as an introduction to this theory. Starting from the basic notions and their properties, the authors pass through the theory of quivers and their representations to the finitely represented algebras. ... Each of fourteen chapters contains a lot of illustrative examples making the book much more readable. The list of 150 articles (or monographs) at the end of the book seems to be sufficiently extensive ..." (European Mathematical Society Newsletter, 1993)"... They (Gabriel and Roiter) are pioneers in this subject and have endeavoured to write this monograph for those people who wish to get involved in the subject. ...The authors paid special attention to the foundations of the subject and they have included proofs for statements which in their opinions are elementary, those which will help further understanding and those which are scarcely available elsewhere. They attempt to take us up to the point where we can find our way in the original literature. ... Any definition which the authors think we have not got to hand is given at the end of each section under the heading "Remarks and References". The reader can soon get used to this cross-referencing. ...Thus as a reference book with basic ideas stated and further reading generously given, it achieves its aims. ..." (The Mathematical Gazette, 1993)"... Written by R.P.Gabriel and A.V.Roiter, with one chapter by B.Keller, it aims at a general outline of old and new results on this subject. Using throughout the language of categories the style of presentation is concise and economical as chosen for an algebraist, who has already some knowledge on that field. ... the standard of this text is high and will be definitely appreciated by the algebraic community." (Monatshefte Mathematik, 1994)"...This book is very welcome because it presents some basic material and at the same time it presents some new insights of the theory. ..." (Zentralblatt für Mathematik, 1996)