The moduli space $M_g$ of genus $g$ curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of $M_g$ is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of $M_g$ called the tautological ring. The definition of the tautological ring was later extended to the compactification $M_g$-bar and the moduli spaces with marked points $M_{g,n}$-bar. While the full cohomology ring of $M_{g,n}$-bar is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I'll ask the question: which cohomology groups $H^k$($M_{g,n}$-bar) are tautological? And when they are not, how can we better understand them? This is joint work with Samir Canning and Sam Payne.