  An important part of the classical theory of real or complex manifolds is the
theory of (smooth, real analytic or complex analytic) vector bundles. With any
vector bundle over a manifold (M,F) the sheaf of its (smooth, real analytic or
complex analytic) sections is associated which is a locally free sheaf of
F-modules, and in this way all the locally free sheaves of F-modules over (M,F)
can be obtained. In the present paper, locally free sheaves of O-modules over a
complex analytic supermanifold (M,O) are studied.
  The main results of the paper are the following ones. Given a locally free
sheaf E of O-modules over a complex analytic supermanifold (M,O), we construct
a locally free sheaf over the retract of (M,O) which is called the retract of
E. Our first result is a classification of locally free sheaves of modules
which have a given retract in terms of non-abelian 1-cohomology. The case of
the tangent sheaf of a split supermanifold is studied in more details. Then we
study locally free sheaves of modules over projective superspaces. A spectral
sequence which connects the cohomology with values in a locally free sheaf of
modules with the cohomology with values in its retract is constructed.
