I will survey recent advances in understanding the unstable cohomology groups of moduli spaces of curves. The approach is fundamentally rooted in motivic structures, such as mixed Hodge theory, and draws inspiration from predictions about l-adic Galois representations of conductor 1 from the automorphic side of the Langlands correspondence. Many of the resulting predictions for the cohomology of moduli spaces of curves are now proved unconditionally, and the appearances of each such representation are governed by a graph complex. By studying these graph complexes, we have described many new infinite families of unstable cohomology groups and obtained arithmetic consequences regarding the nature of the function counting geometric isomorphism classes of curves of fixed genus over varying finite fields.