We study adiabatic limits of Ricci-flat Kähler metrics on a
Calabi-Yau manifold which is the total space of a holomorphic
fibration when the volume of the fibers goes to zero. By establishing some new
a priori estimates for the relevant complex
Monge-Ampère equation, we show that the Ricci-flat metrics collapse
(away from the singular fibers) to a metric on the base of
the fibration. This metric has Ricci curvature equal to a Weil-
Petersson metric that measures the variation of complex structure
of the Calabi-Yau fibers. This generalizes results of Gross-Wilson
for $K3$ surfaces to higher dimensions.