This is the fourth printing of a classic Carus Mathematical Monograph first published in 1968. The book is not a treatise on noncommutative rings. Its aim is to present several fundamental ideas of the theory and some classical theorems. Prerequisities are a first course on abstract algebra and a basic knowledge of field theory as covered in the introductory chapter on fields in van der Waerden’s Modern Algebra. After the first two introductory chapters on the structure of noncommutative rings, which are focused on the Jacobson radical and semisimple rings, the authors cover basic results on commutativity of division rings, simple algebras, representations of finite groups and polynomial identities. The two final chapters present the Goldie theorem and the Golod-Shafarevitch theorem. The book is an excellent introduction to the theory of noncommutative rings for advanced students as well as for non-specialists.