Recognising that the theory of group representations is fundamental to several areas of science and mathematics - including particle physics, crystallography, and group theory - the National Bureau of Standards published this basic but complete exposition of the subject in 1968 in their Applied Mathematics Series. The most significant facts about group representation are developed in an accessible manner, requiring only a familiarity with classical matrix theory. The treatment is rendered self-contained with a series of concise Appendixes that explore elements of the theory of algebraic numbers. Subjects include representations of arbitrary groups, representations of finite groups, multiplication of representations, and bounded representations and Weyl's theorem. All of the important elementary results are featured, a number of advanced topics are discussed, and several special representations are worked out in detail. 1968 edition. AUTHOR: Morris Newman (1924 2007) received his Ph.D. from the University of Pennsylvania and was a research mathematician at the National Bureau of Standards from 1952 77. From 1977 until his 1993 retirement, he was Professor of Mathematics at the University of California, Santa Barbara, where he continued working with students for many years after his retirement. He is the author of Integral Matrices.