Résume  Renormalization is a central idea of contemporary Dynamical Systems
Theory, it allows one to control small scale structure of certain
classes of systems, which leads to universal features of the phase
and parameter spaces. We will review several occurrences of
Renormalization in Holomorphic Dynamics: for quadraticlike, Siegel,
and parabolic maps that enlighten the structure of various dynamical
and parameter images (Julia sets and the Mandelbrot set). In
particular, these ideas helped to construct examples of Julia sets of
positive area (resolving a classical problem going back to Fatou).
First examples were constructed by Buff and Cheritat about 10 years
ago, and more recently a different class, with a number of
interesting new features, was produced by Artur Avila and the author.
In the talk, we will describe these developments.
