The tautological ring of the moduli space of smooth curves of genus g was introduced by Mumford in the 1980s in analogy with the cohomology of Grassmannians. Work of Faber and Faber-Zagier in the 1990s led to two competing conjectural descriptions of the structure of the tautological ring. The two conjectures give two distinct combinatorial characterizations of this ring, and they are both true for genus g = 24. After reviewing these conjectures, I will discuss some of the evidence in recent years favoring one conjecture over the other. Speaker(s): Aaron Pixton (MIT)