1. Introductory example: a gravitational catastrophe machine; 2. Curves, and functions on them; 3. More about functions; 4. Regular values and smooth manifolds; 5. Envelopes; 6. Unfoldings; 7. Unfoldings: applications; 8. Transversality; 9. Generic properties of curves; 10. More on unfoldings; 11. Singular points, several variables and generic surfaces; Appendix: Null sets and Sard's theorem.