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We establish a new and surprisingly strong link between two previously unrelated theories: the theory of moduli spaces of curves M_{g,n} (which, according to Penner, is controlled by the ribbon graph complex) and the homotopy theory of E_{d} 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^{•}(M_{g,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 E_{d} 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 GRT_{1} 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).