Preface; 1. Introduction; Part I. Ordinary Differential Equations: 2. Point transformations and their generators; 3. Lie point symmetries of ordinary differential equations: the basic definitions and properties; 4. How to find the Lie point symmetries of an ordinary differential equation; 5. How to use Lie point symmetries: differential equations with one symmetry; 6. Some basic properties of Lie algebras; 7. How to use Lie point symmetries: second order differential equations admitting a G2; 8. Second order differential equations admitting a G3IX; 9. Higher order differential equations admitting more than one Lie point symmetry; 10 Systems of second order differential equations; 11. Symmetries more general than Lie point symmetries; 12. Dynamical symmetries: the basic definitions and properties; 13. How to find and use dynamical symmetries for systems possessing a Lagrangian; 14. Systems of first order differential equations with a fundamental system of solutions; Part II. Partial Differential Equations: 15. Lie point transformations and symmetries; 16. How to determine the point symmetries of partial differential equations; 17. How to use Lie point symmetries of partial differential equations I: generating solutions by symmetry; 18. How to use Lie point symmetries of partial differential equations II: similarity variables and reduction of the number of variables; 19. How to use Lie point symmetries of partial differential equations III: multiple reduction of variables and differential invariants; 20. Symmetries and the separability of partial differential classification; 21. Contact transformations and contact symmetries of partial differential equations, and how to use them; 22. Differential equations and symmetries in the language of forms; 23. Lie-Backlund transformations; 24. Lie-Backlund symmetries and how to find them; 25. How to use Lie-Backlund symmetries; Appendices; Index.