Phase field models have become popular tools that help capture important features of a variety of physical processes. In this talk, we will briefly discuss two popular models for phase separation: the Allen-Cahn equation and the Chan-Hilliard equation. We will then discuss finite element methods that incorporate continuous data assimilation in order to achieve long time accuracy and stability for arbitrarily inaccurate initial conditions provided enough data measurements are incorporated into the simulation. We will demonstrate the effectiveness of our methods via several numerical experiments.