The notion of graph homology was introduced by Kontsevich in the beginning of nineties. It is related and brings together many different areas such as homological algebra, algebraic geometry, homotopy theory and mathematical physics. The definition is very simple and elementary, however computations are extremely hard; for example the full knowledge of the graph cohomology in the ribbon case is equivalent to the knowledge of the unstable cohomology of the moduli space of complex algebraic curves at any genus.

In my talk I review some new (and not so new) developments in this area.