The Mandelbrot set encodes the dynamical dependence of quadratic polynomials on a parameter. The MLC Conjecture (asserting that the Mandelbrot set is locally connected) is a rigidity property that yields a satisfactory topological description of the Mandelbrot set. In this talk, we describe the historical motivation for the conjecture, explain how it became a central topic in Renormalization Theory (analyzing first-return maps to small neighborhoods of special points), and outline some of the ideas behind the most recent advances toward MLC and related questions.