A systematic introduction to the eigenfunction method, an approach to the group representation theory which was developed by the authors in the 1970’s and 1980’s in accordance with the concept and method used in quantum mechanics. It covers the applications of the group theory in various branches of physics and quantum chemistry, especially nuclear and molecular physics. Extensive tables and computational methods are presented. The volume may serve as a handbook for researchers doing group theory calculations. It can also be used by undergraduate and graduate students who intend to use group theory in their future research careers.