Let the manifold
$X$
parametrise a family of compact complex submanifolds of the complex (or CR) manifold
$Z$
. Under mild conditions the Penrose transform typically provides isomorphisms between a cohomology group of a holomorphic vector bundle
$V\longrightarrow Z$
and the kernel of a differential operator between sections of vector bundles over
$X$
. When the spaces in question are homogeneous for a group
$G$
the Penrose transform provides an intertwining operator between representations.
The paper develops a Penrose transform for compactly supported cohomology on
$Z$
. It provides a number of examples where a compactly supported cohomology group is shown to be isomorphic to the cokernel of a differential operator between compactly supported sections of vector bundles over
$X$
. It considers also how the Serre duality pairing carries through the transform.