  In this paper we study the second integral cohomology of moduli spaces of
semistable sheaves on projective K3 surfaces. If $S$ is a projective K3
surface, $v$ a Mukai vector and $H$ a $v-$generic polarization on $S$, we show
that $H^{2}(M_{v},\mathbb{Z})$ is a free $\mathbb{Z}-$module of rank 23
carrying a pure weight-two Hodge structure and a lattice structure, with
respect to which $H^{2}(M_{v},\mathbb{Z})$ is Hodge isometric to the Hodge
sublattice $v^{\perp}$ of the Mukai lattice of $S$. Similar results are proved
for Abelian surfaces.
