Isometries, conformal and quasiconformal maps

Abstract: 

We give the metric and the analytic definition of different kind of maps in ${\mathbb R}^n$. We sketch a proof of the fact that isometries are affine maps and that conformal maps are Moebius transformations, one that can be extended to metric spaces whose distance may be non-Riemannian. The characterization of isometries in this more general setting was proved by the speaker in collaboration with E. Le Donne. Finally we provide some examples.