I. Basic concepts. - 1. The simplest examples. - 2. The classes ?I. - 3. The quadratic differential of a map. - 4. The local algebra of a map and the Weierstrass preparation theorem. - 5. The local multiplicity of a holomorphic map. - 6. Stability and infinitesimal stability. - 7. The proof of the stability theorem. - 8. Versai deformations. - 9. The classification of stable germs by genotype. - 10. Review of further results. - II. Critical points of smooth functions. - 11. A start to the classification of critical points. - 12. Quasihomogeneous and semiquasihomogeneous singularities. - 13. The classification of quasihomogeneous functions. - 14. Spectral sequences for the reduction to normal forms. - 15. Lists of singularities. - 16. The determinator of singularities. - 17. Real symmetric and boundary singularities. - III. Singularities of caustics and wave fronts. - 18. Lagrangian singularities. - 19. Generating families. - 20. Legendrian singularities. - 21. The classification of Lagrangian and Legendrian singularities. - 22. The bifurcation of caustics and wave fronts. - References. - Further references. Language: English