In the late 70s, the work of Modica, Mortola, De Giorgi, and many others established deep connections between the Allen-Cahn equation, a semi-linear elliptic equation arising in the van der Waals-Cahn-Hilliard theory of phase transitions, and minimal hypersurfaces, i.e. critical points of the area functional. Based on these ideas, in recent years, the combined work of Guaraco, Hutchinson, Tonegawa, and Wickramasekera established the existence of (optimally regular) minimal hypersurfaces in compact manifolds without boundary. In this talk we will consider the Allen-Cahn equation on manifolds with boundary, and describe geometric and analytic aspects of the boundary behavior of the associated limit interfaces. The end goal of this line of investigation is the construction of free boundary minimal hypersurfaces in manifolds with boundary, i.e. submanifolds with vanishing mean curvature and meeting the boundary orthogonally. I will present progress towards this goal, based on joint work with Martin Li and Lorenzo Sarnataro.