  We present certain new properties about the intersection numbers on moduli
spaces of curves $\bar{\sM}_{g,n}$, including a simple explicit formula of
$n$-point functions and several new identities of intersection numbers. In
particular we prove a new identity, which together with a conjectural identity
implies the famous Faber's conjecture about relations in $\mathcal
R^{g-2}(\sM_g)$. These new identities clarify the mysterious constant in
Faber's conjecture and uncover novel combinatorial structures of intersection
numbers.
  We also discuss some numerical properties of Hodge integrals which have
provided numerous inspirations for this work.
