Renormalization theory in Holomorphic Dynamics gives an insight into various self-similarity features of dynamical and parameter fractal sets.

It is also intimately related to such fundamental problems as local connectivity and area of these sets. We will start with an overview of the classical quadratic-like and Siegel Renormalization theories, and will then pass to a recent “Pacman Renormalization Theory” designed to explain observed self-similarity of the Mandelbrot set near its main cardioid. We will then discuss uniform a priori bounds along the main cardioid, and will complete the picture with stating a “Molecule Conjecture”.