  We define the odd symplectic grassmannians and flag manifolds, which are
smooth projective varieties equipped with an action of the odd symplectic group
and generalizing the usual symplectic grassmannians and flag manifolds.
Contrary to the latter, which are the flag manifolds of the symplectic group,
the varieties we introduce are not homogeneous. We argue nevertheless that in
many respects the odd symplectic grassmannians and flag manifolds behave like
homogeneous varieties; in support of this claim, we compute the automorphism
group of the odd symplectic grassmannians, and we prove a Borel-Weil type
theorem for the odd symplectic group.
