It is well known that relations in tautological rings of moduli spaces of curves produce differential equations for generating functions of Gromov-Witten invariants for all compact symplectic manifolds. We call such equations universal equations. These equations are very helpful in understanding structures of Gromov-Witten theory and played an important role in the so called Virasoro conjecture. However for finding explcit universal equations seems to be a hard problem especially when genus is large. I will talk about a genus-3 topological recursion relation found together with T. Kimura and some recently discovered universal equations conjectured by K. Liu and H. Xu and proved in a joint work with R. Pandharipande.