In the talk I will prove that the moduli spaces M_g,n of n-marked curves of any genus g > 4 do not admit holomorphic 1-forms. The main difficulty is to prove the result for M_g, then one concludes the other cases by an inductive argument. This sheds new light on the question of how far M_g,n is from being projective. The work is joint with F.F. Favale and G.P.Pirola.