This volume contains the proceedings of two related workshops held at The Fields Institute in February and March 1993. The workshops were an integral part of the thematic year in Dynamical Systems and Bifurcation Theory held during the 1992-1993 academic year. This volume covers the full spectrum of research involved in combining symmetry methods with dynamical systems and bifurcation theory, from the development of the mathematical theory in order to understand the underlying mechanisms to the application of this new mathematical theory, to partial differential equation models of realistic physical phenomena. The individual contributions show the richness of the interplay between abstract methods and significant, essential examples. This book features paper by Mike Field summarizing the content of the mini-course that he presented on the geometric analysis of symmetry breaking for compact Lie groups.