Homotopy type of the moduli space of stable rational curves

I shall show that the rational cohomology of the moduli space of stable rational curves is a Koszul algebra (answering a question of Yu. I. Manin, D. Petersen and V. Reiner), and explain how this allows one to compute the rational homotopy invariants of this space in a very explicit way. Time permitting, I shall talk about a few classes of spaces for which similar results are available, and a few other conjectural classes of spaces like that.