The class of stable curves is deformation-open and satisfies the unique limit property, hence gives rise to the modular Deligne-Mumford compactification of $M_{g,n}$. But the class of stable curves is not unique in this respect; one obtains alternate compactifications by considering, for example, a moduli problem in which elliptic tails are replaced by cusps or in which marked points are allowed to collide. In this talk, we will survey progress toward a systematic classification of these alternate compactifications.