  This article is a contribution to the understanding of the geometry of the
twistor space of a symplectic manifold. We consider the bundle $Z$ with fibre
the Siegel domain Sp(2n,R)/U(n) existing over any given symplectic 2n-manifold
M. Then, after recalling the construction of the almost complex structure
induced on $Z$ by a symplectic connection on M, we study and find some specific
properties of both. We show a few examples of twistor spaces, develop the
interplay with the symplectomorphisms of M, find some results about a natural
almost Hermitian structure on $Z$ and finally prove its n+1-holomorphic
completeness. We end by proving a vanishing theorem about the Penrose
transform.
