let m\_g be the moduli space of smooth curves of genus g. the tautological ring is a subring of the cohomology of m\_g that 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. after reviewing these conjectures, i will discuss some of the evidence in recent years favoring one conjecture over the other.