  The goal is to verify the Hodge conjecture (and some related conjectures) for
certain moduli spaces. It is shown that the (generalized) Hodge conjecture
holds for the projective moduli spaces of vector bundles over an abelian or K3
surface or a curve if it holds for all powers of the surface or curve. In the
latter case, the curve can be replaced by its Jacobian. As a consequence the
generalized Hodge conjecture holds for these moduli spaces when the curve is
very general. A similar result is proved for the Hilbert schemes of points over
an arbitrary surface.
