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.