This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). Group theoretical methods are a powerful tool both in their applications to mathematics and to physics. The broad goal of this project was to use such methods to develop the implications of group (symmetry) structures underlying models of physical systems, as well as to broaden the understanding of simple models of chaotic systems. The main thrust was to develop further the complex mathematics that enters into many-particle quantum systems with special emphasis on the new directions in applied mathematics that have emerged and continue to surface in these studies. In this area, significant advances in understanding the role of SU(2) 3nj-coefficients in SU(3) theory have been made and in using combinatoric techniques in the study of generalized Schur functions, discovered during this project. In the context of chaos, the study of maps of the interval and the associated theory of words has led to significant discoveries in Galois group theory, to the classification of fixed points, and to the solution of a problem in the classification of DNA sequences.