  We show that the integral cohomology rings of the moduli spaces of stable
rational marked curves are Koszul. This answers an open question of Manin.
Using the machinery of Koszul spaces developed by Berglund, we compute the
rational homotopy Lie algebras of those spaces, and obtain some estimates for
Betti numbers of their free loop spaces in case of torsion coefficients. We
also prove and conjecture some generalisations of our main result.
