Explore the intricacies of moduli spaces of curves in this colloquium talk by Hannah Larson from Berkeley. Delve into the cohomology of M_g, the moduli space of genus g curves, and its significance in algebraic geometry and related fields. Learn about Mumford's tautological ring, its extension to compactified moduli spaces, and the challenges in understanding the full cohomology ring of M_{g,n}-bar. Investigate the question of which cohomology groups H^k(M_{g,n}-bar) are tautological and how to approach non-tautological cases. Gain insights into joint research with Samir Canning and Sam Payne on this fundamental topic in mathematics.