In addition to the Deligne-Mumford compactification for the moduli space of genus four curves, there are a number of additional compactifications that arise naturally. In this talk I will discuss joint work with Radu Laza where we compare some of these spaces. The description we obtain is similar to that for genus three curves (work of Hyeon-Lee), as well as to some previous results we have for the moduli space of cubic threefolds.