This textbook gives a comprehensive review of the new approach to group representation theory developed in the mid 70's and 80's. The unique feature of the approach is that it is based on Dirac's complete set of commuting operators theory in quantum mechanics and thus the representation theories for finite groups, infinite discrete groups and Lie groups are all unified. The new approach is easily acceptable to physicists, yet powerful and versatile for practical calculations. The development of the theory is from special to general, supplemented with abundant illustrative examples. The Clebsch-Gordan, Racah and subduction coefficients and isoscalar factors are discussed in detail for point groups, permutation groups, unitary groups and space groups. Tables of several useful coefficients are included. The theory on roots and weights in Lie groups is reformulated in the spirit of representation theory of quantum mechanics. The applications of group theory to many-body problem are introduced with emphasis on the various dynamic symmetry models of nuclei. There should be no difficulty for the readers of this book to understand the conclusions derived in other textbooks or literature, although the process of derivation may be entirely different from the conventional one.