Abstract: Northcott proved in 1950 that if $K$ is a number field (i.e., a finite extension of the field of rational numbers) and if $f$ is a rational function of degree at least $2$ defined over $K$, then $f$ has only finitely many preperiodic points defined over $K$. In the 1990's, Morton and Silverman conjectured that the number of such points is bounded above by a constant depending only on the degrees $\deg(f)$ and $[K : Q]$. We will discuss progress on the case when $f$ is a quadratic polynomial and $[K : Q]$ is equal to $1$ or $2$. The main idea will be to associate to the family of quadratic polynomials certain "dynamical modular curves", which are moduli spaces whose points correspond to (equivalence classes of) tuples $(f, P_1, ..., P_n)$ of a quadratic polynomial $f$ and a collection of marked points $P_1, \ldots, P_n$ that satisfy a specified dynamical relation under iteration of $f$. We then discuss progress on the problem of determining the full set of rational or quadratic points on such curves.