1 Introduction

2 Preliminaries

3 Density of Hyperbolicity, Structural Stability and the Bifurcation Locus

4 The Mandelbrot Local Connectivity Conjecture (MLC), Local Connectivity of Julia Sets and topological models

5 Topological rigidity and Quasiconformal rigidity

6 The No Invariant Line Fields Conjecture (NILF)

7 Renormalization and Branner-Hubbard-Yoccoz Puzzle

8 Combinatorial Rigidity and Triviality of Fibers