1. Algebraic functions -- 2. Riemann surfaces -- 3. The sheaf of germs of holomorphic functions -- 4. The Riemann surface of an algebraic function -- 5. Sheaves -- 6. Vector bundles, line bundles and divisors -- 7. Finiteness theorems -- 8. The Dolbeault isomorphism -- 9. Weylโ{128}{153}s lemma and the Serre duality theorem -- 10. The Riemann-Roch theorem and some applications -- 11. Further properties of compact Riemann surfaces -- 12. Hyperelliptic curves and the canonical map -- 13. Some geometry of curves in projective space -- 14. Bilinear relations -- 15. The Jacobian and Abelโ{128}{153}s theorem -- 16. The Riemann theta function -- 17. The theta divisor -- 18. Torelliโ{128}{153}s theorem -- 19. Riemannโ{128}{153}s theorem on the singularities of ?