We survey recent results on algebraic cycles on the moduli space of principally polarized abelian varieties. In particular, we will discuss the tautological ring, the decomposition of cycle classes into tautological and non-tautological components via the tautological projection, and explicit examples of non-tautological classes. A parallel theory can be developed for the universal abelian variety. This work is based on joint projects with Samir Canning, Lycka Drakengren, Jeremy Feusi, Daniel Holmes, Aitor Iribar Lopez, Denis Nesterov, Rahul Pandharipande, Johannes Schmitt, and Zheming Sun.