This book serves as an introduction to the use of nonlinear symmetries in studying, simplifying and solving nonlinear equations. Part I provides a self-contained introduction to the theory. This emphasizes an intuitive understanding of jet spaces and the geometry of differential equations, and a special treatment of evolution problems and dynamical systems, including original results. In Part II the theory is applied to equivariant dynamics, to bifurcation theory and to gauge symmetries, reporting recent results by the author. In particular, the fundamental results of equivariant bifurcation theory are extended to the case of nonlinear symmetries. The final part of the book gives an overview of new developments, including a number of applications, mainly in the physical sciences. An extensive and up-to-date list of references dealing with nonlinear symmetries completes the volume.
This volume will be of interest to researchers in mathematics and mathematical physics.

List of abbreviations. Foreword. Introduction. I: Geometric setting. a) Equations and functions as geometrical objects. b) Symmetry. II: Symmetries and their use. III: Examples. IV: Evolution equations. a) Evolution equations -- general features. b) Dynamical systems (ODEs). c) Periodic solutions. d) Evolution PDEs. V: Variational problems. VI: Bifurcation problems. VII: Gauge theories. VIII: Reduction and equivariant branching lemma. IX: Further developments. X: Equations of physics. References and bibliography. Subject Index.