In 1998, Poonen gave a conditional classification of the rational preperiodic points for quadratic polynomials defined over Q – a dynamical analogue of Mazur's theorem for torsion points on elliptic curves. In an ongoing project (with Xander Faber, David Krumm, and Joe Wetherell), we are attempting to give a similar classification over all quadratic extensions of Q. In this talk, I will describe the progress that has been made, as well as what remains in order to make such a classification.