The intersection theory on moduli spaces of curves is connected to KdV hierarchy through the celebrated Witten-Kontsevich theorem. There is much recent progress on Faber's remarkable conjectures about the structure of tautological rings of moduli spaces of curves, especially their Gorensteiness. The subject is closely related to Gromov-Witten invariants, Weil-Petersson volumes, integrable systems and matrix integrals (Eynard-Orantin theory).