  Based on a recent extension theorem for reflexive differential forms, that
is, regular differential forms defined on the smooth locus of a possibly
singular variety, we study the geometry and cohomology of sheaves of reflexive
differentials.
  First, we generalise the extension theorem to holomorphic forms on locally
algebraic complex spaces. We investigate the (non-)existence of reflexive
pluri-differentials on singular rationally connected varieties, using a
semistability analysis with respect to movable curve classes. The necessary
foundational material concerning this stability notion is developed in an
appendix to the paper. Moreover, we prove that Kodaira-Akizuki-Nakano vanishing
for sheaves of reflexive differentials holds in certain extreme cases, and that
it fails in general. Finally, topological and Hodge-theoretic properties of
reflexive differentials are explored.
