The cohomology of the moduli space of stable curves has been widely studied, but in general, understanding the full cohomology ring of this space is too much to ask. Instead, one generally settles for studying the tautological ring, a subring of the cohomology that is simultaneously tractable to study and yet rich enough to contain most cohomology classes of geometric interest. The first known example of an algebraic cohomology class that is not tautological was discovered by Graber and Pandharipande, in work that was later significantly generalized by van Zelm to produce an infinite family of non-tautological classes on the moduli space of stable curves. A similar study can be undertaken on the moduli space of smooth curves, but in this case, almost no non-tautological classes were previously known. I will report on joint work with V. Arena, S. Canning, R. Haburcak, A. Li, S.C. Mok, and C. Tamborini (from the 2023 AGNES Summer School), in which we produce non-tautological algebraic classes on the moduli space of smooth curves in an infinite family of cases, including on M_g for all g>15.