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As was shown by Harer the second homology of Mg, the moduli space of compact Riemann surfaces of genusg, is of rank 1, provided that g ≥ 3. This means that a nontrivial second de Rham cohomology class on Mg is unique up to a constant factor. But several canonical 2-forms on the moduli space have been constructed in various geometric contexts and differ from each other. In this chapter we review some of constructions in order to provide material for future research on “secondary geometry” of the moduli space Mg.