The Deligne-Mumford compactification of the moduli space of smooth Riemann surfaces is obtained by allowing nodal Riemann surfaces. In various questions in Teichmuller dynamics and algebraic geometry, it is natural to consider also degenerations of a family of Riemann surfaces, each equipped with a suitable meromorphic differential. The main difficulty in compactifying such families is that in the limit the differential may become identically zero on some components of the nodal curve. We will describe recent joint works with M. Bainbridge, D. Chen, Q. Gendron, M. Moeller, and with I. Krichever and C. Norton, which aim at constructing compactifications preserving more information about degenerations of differentials, and at describing the limits geometrically.