1．Algebraic functions 3
2．Riemann surfaces 8
3．The sheaf of germs of holomorphic functions 12
4．The Riemann surface of an algebraic function 15
5．Sheaves 17
6．Vector bundles,line bundles and divisors 27
7．Finiteness theorems 32
8．The Dolbeault isomorphism 38
9．Weyl's lemma and the Serre duality theorem 43
10．The Riemann-Roch theorem and some applications 49
11．Further properties of compact Riemann surfaces 58
12．Hyperelliptic curves and the canonical map 63
13．Some geometry of curves in projective space 66
14．Bilinear relations 77
15．The Jacobian and Abel's theorem 84
16．The Riemann theta function 91
17．The theta divisor 97
18．Torelli's theorem 106
19．Riemann's theorem on the singularities of θ 111
References 119