We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves Mg,n (which, according to Penner, is controlled by the ribbon graph complex) and the homotopy theory of Ed operads (controlled by ordinary graph complexes with no ribbon structure, introduced first by Kontsevich). The link between the two goes through a new intermediate {\em stable}\, ribbon graph complex which has roots in the deformation theory of quantum A∞ algebras and the theory of Kontsevich compactifications of moduli spaces of curves \overline{\mathcal M}_{g,n}^K. Using a new prop of ribbon graphs and the fact that it contains the prop of involutive Lie bialgebras as a subprop we find new algebraic structures on the classical ribbon graph complex computing H•(Mg,n). We use them to prove Comparison Theorems, and in particular to construct a non-trivial map from the ordinary to the ribbon graph cohomology. On the technical side, we construct a functor O from the category of prop(erad)s to the category of operads. If a properad P is in addition equipped with a map from the properad governing Lie bialgebras (or graded versions thereof), then we define a notion of P-``graph'' complex, of stable P-graph complex and a certain operad, that is in good cases an Ed operad. In the ribbon case, this latter operad acts on the deformation complexes of any quantum A∞-algebra. We also prove that there is a highly non-trivial, in general, action of the Grothendieck-Teichm\"uller group GRT1 on the space of so-called {\em non-commutative Poisson structures}\, on any vector space W equipped with a degree −1 symplectic form (which interpolate between cyclic A∞ structures in W and ordinary polynomial Poisson structures on W as an affine space).
