Abstract: There are a variety of choices for compactifying the moduli space of curves, the most well-known of which is the Deligne-Mumford-Knudsen compactification by stable curves. Several alternative compactifications have been constructed and studied since, and in 2013 David Smyth used a combinatorial framework to make progress towards classifying all “sufficiently nice” compactifications. In this talk, I’ll discuss Smyth’s framework and present two new families of compactifications, which together classify all Gorenstein compactifications in genus 0 and genus 1. This is based on joint work with Sebastian Bozlee.