While studying two Riemann surfaces it is natural to consider them equivalent if there exists a holomorphic homeomorphism between them. The set of such equivalence classes is the moduli or Riemann space. It was Teichmüller who realized that introducing a new relation, based on quasiconformal mappings (and some homotopy conditions) one gets a space, known as the Teichmüller space, which is simpler to study than the Riemann space. This new space is the universal covering space of Riemann space; the mapping class group (see Nielsen's theorem 1.33) becomes the covering group. Thus one reduces the study of Riemann space to the study of Teichmüller space and the mapping class group.

In the first two subsections we study some examples of classification of certain Riemann surfaces where it is possible to do computations explicitly. The rest of the subsections are devoted to the definition and properties of Teichmüller spaces of Riemann surfaces and Fuchsian groups. The basic material of this section is taken from S. Nag's book [39]; other good references are the books by Abikoff [1] (for the real analytic theory), Gardiner [20] and Y. Imayoshi and M. Taniguchi [26].

- The Riemann Mapping Theorem and the Uniformization Theorem
- Abel's theorem and the uniformization of tori
- Teichmüller spaces of Riemann surfaces, moduli spaces and modular groups
- Teichmüller spaces of Fuchsian groups
- Fricke coordinates
- Teichmüller's theorem
- Teichmüller's lemma and the complex structure of Teichmüller spaces
- Bers embedding and the Bers boundary of Teichmüller space
- Royden's Theorems
- The Patterson and Bers-Greenberg Isomorphism Theorems
- The Weil-Petersson metric