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The famous Witten's conjecture, proposed in 1991, claims that the generating series of intersection numbers on the moduli space of stable algebraic curves gives a solution of the Korteweg - de Vries hierarchy. Since that time, a lot of results, showing a close relation of the topology of the moduli space of curves with the theory of partial differential equations, possessing an infinitesimal number of symmetries, were obtained. In my talk, I will recall some of these results and will present a general construction that, given a system of cohomology classes on the moduli spaces of curves, satisfying a collection of simple properties, associates to it a system of partial differential equations with a rich algebraic structure. This construction gives a new information about generalizations of Witten's conjecture and also, unexpectedly, predicts new relations in the cohomology of the moduli space of curves.