I am about to teach a first course on Riemann surfaces, and I am trying to get a fairly comprehensive view of the main references, as a support for both myself and students.
I compiled a list, here goes in alphabetical order. Of course, it is necessarily subjective. For more detailed entries, I made a bibliography using the bibtex entries from MathSciNet: click here.
- Bobenko. Introduction to compact Riemann surfaces.
- Bost. Introduction to compact Riemann surfaces, Jacobians, and abelian varieties.
- de Saint-Gervais. Uniformisation des surfaces de Riemann: retour sur un théorème centenaire.
- Donaldson. Riemann surfaces.
- Farkas and Kra. Riemann surfaces.
- Forster. Lectures on Riemann surfaces.
- Griffiths. Introduction to algebraic curves.
- Gunning. Lectures on Riemann surfaces.
- Jost. Compact Riemann surfaces.
- Kirwan. Complex algebraic curves.
- McMullen. Complex analysis on Riemann surfaces.
- McMullen. Riemann surfaces, dynamics and geometry.
- Miranda. Algebraic curves and Riemann surfaces.
- Narasimhan. Compact Riemann surfaces.
- Narasimhan and Nievergelt. Complex analysis in one variable.
- Reyssat. Quelques aspects des surfaces de Riemann.
- Springer. Introduction to Riemann surfaces.
- Varolin. Riemann surfaces by way of complex analytic geometry.
- Weyl. The concept of a Riemann surface.
Having a good sense of what each of these books does, beyond a superficial first impression, is quite a colossal task (at least for me).
What I'm hoping is that if you know very well such or such reference in the list, you can give a short description of it: where it stands in the existing literature, what approach/viewpoint is adopted, what are its benefits and pitfalls. Of course, I am also happy to update the list with new references, especially if I missed some major ones.
It is extremely well-written, but definitely more analytic in flavor. In particular, it includes pretty much all the analysis to prove finite-dimensionality of sheaf cohomology on a compact Riemann surface. It also deals quite a bit with non-compact Riemann surfaces, but does include standard material on Abel's Theorem, the Abel-Jacobi map, etc.