  The renormalization of a quadratic-like map is studied. The three-dimensional
Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed.
For an unbranched quadratic-like map having the {\sl a priori} complex bounds,
the local connectivity of its Julia set is proved by using the
three-dimensional Yoccoz puzzle. The generalized version of Sullivan's sector
theorem is discussed and is used to prove his result that the Feigenbaum
quadratic polynomial has the {\sl a priori} complex bounds and is unbranched. A
dense subset on the boundary of the Mandelbrot set is constructed so that for
every point of the subset, the corresponding quadratic polynomial is unbranched
and has the {\sl a priori} complex bounds.
