  In this paper, first we modify the definition of a Hom-Lie algebroid
introduced by Laurent-Gengoux and Teles and give its equivalent dual
description. Many results that parallel to Lie algebroids are given. In
particular, we give the notion of a Hom-Poisson manifold and show that there is
a Hom-Lie algebroid structure on the pullback of the cotangent bundle of a
Hom-Poisson manifold. Then we give the notion of a Hom-Lie bialgebroid, which
is a natural generalization of a purely Hom-Lie bialgebra and a Lie
bialgebroid. We show that the base manifold of a Hom-Lie bialgebroid is a
Hom-Poisson manifold. Finally, we introduce the notion of a Hom-Courant
algebroid and show that the double of a Hom-Lie bialgebroid is a Hom-Courant
algebroid. The underlying algebraic structure of a Hom-Courant algebroid is a
Hom-Leibniz algebra, or a Hom-Lie 2-algebra.
