Dynamics in the family $e^{2\pi i \theta} z + z^2$ is one of the most
delicate themes of Holomorphic Dynamics. Three regimes, parabolic, Siegel
and Cremer, are intertwined in an intriate way depending on the
Diophantine properties of the rotation number $\theta$. Global structure of
Siegel maps of bounded type is now well understood due to the methods of
quasiconformal surgery and renormalization. On the other hand, the
structure of maps of high type is also well understood by means of
parabolic renormalization. Recently new methods of Near Degenerate Regime
have been developed giving a chance for complete understanding of this
family. In the talk we will give a self-contained overview of this story.