Conformal dynamics classically concerns the dynamics of Kleinian groups and of rational maps on the Riemann sphere.
Many methods turn out to be rather topological and extend to a much broader setting, e.g. word-hyperbolic groups acting on their visual boundaries and finite branched coverings acting on some repellors.

The aim of this talk will be to introduce these objects and to present recent progress on the topological characterizations
of Kleinian groups and rational maps.