These notes were written for some expository talks on Teichmüller theory, given
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.