  We study a modification of Poisson geometry by a closed 3-form. Just as for
ordinary Poisson structures, these "twisted" Poisson structures are
conveniently described as Dirac structures in suitable Courant algebroids. The
additive group of 2-forms acts on twisted Poisson structures and permits them
to be seen as glued from ordinary Poisson structures defined on local patches.
We conclude with remarks on deformation quantization and twisted symplectic
groupoids.
