The tautological ring of the moduli space of curves of genus g with n markings is a certain subalgebra of its cohomology/Chow ring generated by particularly natural cycle classes. A fruitful way of studying the tautological ring is by considering the Leray spectral sequence for the morphism that forgets the marked points; this is particularly true in low genus, where the spectral sequence is strongly related to automorphic forms. I will explain how this has led to a precise description of the tautological ring in genus one, a series of counterexamples to the Faber conjectures in genus two (partially joint work with Orsola Tommasi), and if time permits, some new results in genus three (joint with Mehdi Tavakol).