  Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory
are studied. We find a universal system of differential equations which
determines the generating function of these integrals from the standard
descendent potential (for any target X). We use virtual localization and
classical degeneracy calculations to find trigonometric closed form solutions
for special Hodge integrals over the moduli space of pointed curves. These
formulas are applied to two computations in Gromov-Witten theory for Calabi-Yau
3-folds. The genus g, degree d multiple cover contribution of a rational curve
is found to be simply proportional to the Euler characteristic of M_g. The
genus g, degree 0 Gromov-Witten invariant is calculated (in agreement with
recent string theoretic calculations of Gopakumar-Vafa and Marino-Moore).
Finally, with Zagier's help, our Hodge integral formulas imply a general genus
prediction of the punctual Virasoro constraints applied to the projective line.
