  Generalized geometry finds many applications in the mathematical description
of some aspects of string theory. In a nutshell, it explores various structures
on a generalized tangent bundle associated to a given manifold. In particular,
several integrability conditions can be formulated in terms of a canonical
Dorfman bracket, an example of Courant algebroid. On the other hand, smooth
manifolds can be generalized to involve functions of $\mathbb{Z}$-graded
variables which do not necessarily commute. This leads to a mathematical theory
of graded manifolds. It is only natural to combine the two theories by
exploring the structures on a generalized tangent bundle associated to a given
graded manifold.
  After recalling elementary graded geometry, graded Courant algebroids on
graded vector bundles are introduced. We show that there is a canonical bracket
on a generalized tangent bundle associated to a graded manifold. Graded
analogues of Dirac structures and generalized complex structures are explored.
We introduce differential graded Courant algebroids which can be viewed as a
generalization of Q-manifolds. A definition and examples of graded Lie
bialgebroids are given.
