We present applications of the group theory to the solution and systematization of several problems in the theory of differential equations (for instance, a base for the method of separation of variables), classical mechanics (for instance, the analytic form of the Lagrangian), relativity theory, quantum mechanics, and elementary particle physics. By this comprehensive work we wish to give the reader a number of preliminary notions and examples which are absolutely necessary for a better understanding of certain works, in different domains, which are based on applications of group theory. The present volume is a new edition of a volume published in 1985, ( Aplicatii ale teoriei grupurilor in mecanica si fizica , Editura Tehnica, Bucharest, Romania). This new edition contains many improvements concerning the presentation, as well as new topics using an enlarged and updated bibliography. In addition to the large area of domains in physics covered by this volume, we are presenting both discrete and continuous groups, while most of the books about applications of group theory in physics present only one type of groups (i.e., discrete or continuous), and the number of analyzed groups is also relatively small (i. e., point groups of crystallography, or the groups of rotations and translations as examples of continuous groups; some very specialized books study the Lorentz and Poincare groups of relativity theory). The work requires as preliminaries only the mathematical knowledge acquired by a student in a technical university. It is addressed to a large audience, to all those interested and compelled to use mathematical methods in various fields of research, such as mechanics, physics, engineering, people involved in research or teaching, as well as students.