The theory of algebraic curves is essentially equivalent to the theory of algebraic function fields. The latter requires less background and is closer to the theory of finite fields; therefore we present here the theory of function fields. At the end of the section, we give a brief introduction to the language of algebraic curves. Our exposition follows mainly the book [2714]∗, other references are [1147, 1296, 1511, 2280, 2281, 2872].