  Motivated by the Strominger-Yau-Zaslow conjecture, we study fibre spaces
whose total space has trivial canonical bundle. Especially, we are interest in
Calabi-Yau varieties with fibre structure. In this paper, we only consider
semi-stable families. We use Hodge theory and the generalized
Donaldson-Simpson-Uhlenbeck-Yau correspondence to study the parabolic structure
of higher direct images over higher dimensional quasi-projective base, and
obtain an important result on parabolic-semi-positivity. We then apply this
result to study nonisotrivial Calabi-Yau varieties fibred by Abelian varieties
(or fibred by hyperk\"ahler varieties), we obtain that the base manifold for
such a family is rationally connected and the dimension of a general fibre
depends only on the base manifold.
