  In this paper, we introduce the notion of $E$-Courant algebroids, where $E$
is a vector bundle. It is a kind of generalized Courant algebroid and contains
Courant algebroids, Courant-Jacobi algebroids and omni-Lie algebroids as its
special cases. We explore novel phenomena exhibited by $E$-Courant algebroids
and provide many examples. We study the automorphism groups of omni-Lie
algebroids and classify the isomorphism classes of exact $E$-Courant
algebroids. In addition, we introduce the concepts of $E$-Lie bialgebroids and
Manin triples.
