In 1990--1998, my main focus was the dynamics in the real quadratic family,
which I approached by complex methods.
The main results are:
No wild attractors  (solution of Milnor's Problem in the quadratic case);
Density of hyperbolic maps ;
Proof of the Feigenbaum-Coullet-Tresser Renormalization Conjecture ;
Typical non-regular and non-renormalizable maps are stochastic
(satisfying the Martens-Nowicki criterion) ;
Construction of the Full Renormalization Horseshoe .
The last two papers imply the Regular or Stochastic Theorem :
Almost any real quadratic map is either regular or stochastic.
Note  is a survey on the subject based on my plenary talk at the AMS meeting in
Washington, DC (January 2000).