  In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of
$h$-polarized hyperk\"ahler manifolds. Firstly, we construct the tautological
ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the
tautological conjecture on moduli spaces of K3 surfaces. We study the
tautological classes in cohomology groups and prove that most of them are
linear combinations of Noether-Lefschetz cycle classes. In particular, we prove
the cohomological version of the tautological conjecture on moduli space of
K3$^{[n]}$-type hyperk\"ahler manifolds with $n\leq 2$. Secondly, we prove the
cohomological generalized Franchetta conjecture on universal family of these
hyperk\"ahler manifolds.
