Abstract: In the paper "Formal noncommutative symplectic geometry'', Maxim Kontsevich introduced three versions of cochain complexes GCCom, GCLie and GCAs "assembled from'' graphs with some additional structures. The graph complex GCCom (resp. GCLie, GCAs) is related to the operad Com (resp. Lie, As) governing commutative (resp. Lie, associative) algebras. Although the graphs complexes GCCom, GCLie and GCAs (and their generalizations) are easy to define, it is hard to get very much information about their cohomology spaces. In my talk, I will describe the links between these graph complexes (and their modifications) to the cohomology of the moduli spaces of curves, the group of outer automorphisms Out(Fr) of the free group Fr on r generators, the absolute Galois group Gal(Q¯/Q) of rationals, finite type invariants of tangles, and the homotopy groups of embedding spaces.