  As was shown by Harer the second homology of ${\mathbb M}_g$, the moduli
space of compact Riemann surfaces of genus $g$, is of rank 1, provided $g \geq
3$. This means a nontrivial second de Rham cohomology class on ${\mathbb M}_g$
is unique up to 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 article we review some of constructions in order to provide
materials for future research on "secondary geometry" of the moduli space
${\mathbb M}_g$.
