Deformations of complex superspaces and of the coherent sheaves on them

This paper sets forth the basic elements of the theory of complex superspaces, coherent sheaves on them and deformations of these objects. The existence of versal deformations is proved for various objects of superanalytical geometry — superspaces, sheaves on them, subsuperspaces, SUSY-curves and so on. A construction is described of families of supermanifolds that contain all supermanifolds with a given underlying or associated split manifold.