Shimura varieties are algebraic varieties defined from group-theoretic data whose cohomology is a useful tool in the Langlands programme. In this talk I will discuss the 'Tautological Ring' of a Shimura variety, which is a natural invariant that encodes refined geometric information. I'll present a conjecture which describes the tautological ring directly from the initial group-theoretic data and describe some recent work making progress towards proving the conjecture (joint with Wushi Goldring). The conjecture provides a concrete presentation of the tautological ring. I will present this in the case of the modular curve and, more generally, the Siegel modular variety where the conjecture is known to be true.