By the work of Yoccoz the local connectivity of the Mandelbrot set reduces to showing that there is a unique infinitely renormalizable map with a given combinatorics; this is typically shown by controlling the geometry of any such map. I will describe the work of Misha on "high type" renormalization and my own work on "bounded-primitive type", and then explain how these results were combined to give the "finite decorations" and "away from the central molecule" results. If time permits I will state our latest result, on the "eye of the elephants", and speculate as to the final structure of the proof of MLC.