The Mandelbrot set M captures in one picture tremendous dynamical complexity of the quadratic family z^2+c. One of its remarkable features is that it contains small copies of itself almost indistinguishable from the whole set. The quadratic-like renormalization theory is designed to explain this phenomenon. It would imply the MLC Conjecture on local connectivity of M, which in turn would provide us with a precise topological model for this fractal set. In the talk (that will start from skratch) we will update the audience on the current status of these problems. In particular, only recently the MLC was established at the

Feigenbaum parameter born through the cascade of doubling bifurcatons that was the last problematic real point.