The study of interconnections between integrable systems and topology of Deligne - Mumford moduli spaces of stable algebraic curves and their generalisations has been initiated by E.Witten and M.Kontsevich at the beginning of 90s. In the talk I will explain some basic ideas about the conjectural correspondence between the class of smooth projective varieties with semisimple quantum cohomology and a class of hierarchies of integrable PDEs.