  We describe how generalized complex geometry, which interpolates between
complex and symplectic geometry, is compatible with T-duality, a relation
between quantum field theories discovered by physicists. T-duality relates
topologically distinct torus bundles, and prescribes a method for transporting
geometrical structures between them. We describe how this relation may be
understood as a Courant algebroid isomorphism between the spaces in question.
This then allows us to transport Dirac structures, generalized Riemannian
metrics, generalized complex and generalized Kahler structures, extending the
"Buscher rules" well-known to physicists. Finally, we re-interpret T-duality as
a Courant reduction, and explain that T-duality between generalized complex
manifolds may be viewed as a generalized complex submanifold (D-brane) of the
product, in a way that establishes a direct analogy with the Fourier-Mukai
transform.
