These notes were written for some expository talks on Teichmüller theory, given at Conference on Low-dimensional manifolds and Groups , held in the Indian Statistical Institute, Bangalore, July 12-24, 2004. I have tried to make §§3 and 4 a little independent from §§1 and 2 (if one has a little background in the theory) and of each other. So some definitions might appear in §3 and also in §4.

The notes are far from complete, proofs are not given; they are meant as a guide for lecturing, rather than a formal introduction to the theory of Teichmüller spaces.

Pablo Ares Gastesi 2005-08-31