Dynamics of quadratic polynomials over quadratic fields

In 1998, Poonen gave a conjecturally complete classification of the possible preperiodic structures for quadratic polynomials defined over Q. In this thesis, we prove a number of results toward a similar classification, but over quadratic extensions of Q. In order to do so, we formalize a more general notion of emph{dynamical modular curve} in the setting of quadratic polynomial dynamics. We also discuss the geometry of these curves, proving in some cases that they are irreducible over C, extending a result of Bousch.