  This is an introduction to the geometry of compact Riemann surfaces, largely
following the books Farkas-Kra, Fay, Mumford Tata lectures. 1) Defining Riemann
surfaces with atlases of charts, and as locus of solutions of algebraic
equations. 2) Space of meromorphic functions and forms, we classify them with
the Newton polygon. 3) Abel map, the Jacobian and Theta functions. 4) The
Riemann--Roch theorem that computes the dimension of spaces of functions and
forms with given orders of poles and zeros. 5) The moduli space of Riemann
surfaces, with its combinatorial representation as Strebel graphs, and also
with the uniformization theorem that maps Riemann surfaces to hyperbolic
surfaces. 6) An application of Riemann surfaces to integrable systems, more
precisely finding sections of an eigenvector bundle over a Riemann surface,
which is known as the "algebraic reconstruction" method in integrable systems,
and we mention how it is related to Baker-Akhiezer functions and Tau functions.
