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 (or Chow) ring of \M_{g,n}-bar is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I will address the question: for which g and n does the tautological subring coincide with the full cohomology (or Chow) ring? In joint work with Samir Canning, we answer several new cases of this question.