1. Introduction -- 2. Actions on functions, difficulties -- 3. Parametric representation of functions -- 4. Action on parametric representations -- 5. Parametric functions as solutions -- 6. Rarefaction waves and Riemann solvers of the nonlinear shock wave equation -- 7. Arbitrary nonlinear Lie group actions on generalised functions -- 8. Nonprojectable Lie group symmetries of rarefaction waves and Riemann solvers -- 9. General parametric approach to Lie symmetries -- 10. Projectable Lie group actions and Hilbert's fifth problem -- 11. Nonprojectable Lie group actions and an answer to Hilbert's fifth problem -- 12. Singularities and the nowhere dense algebras of generalised functions -- 13. Lie semigroup actions and semisymmetries.