  Holonomy groups and holonomy algebras for connections on locally free sheaves
over supermanifolds are introduced. A one-to-one correspondence between
parallel sections and holonomy-invariant vectors, and a one-to-one
correspondence between parallel locally direct subsheaves and
holonomy-invariant vector supersubspaces are obtained. As the special case, the
holonomy of linear connections on supermanifolds is studied. Examples of
parallel geometric structures on supermanifolds and the corresponding
holonomies are given. For Riemannian supermanifolds an analog of the Wu theorem
is proved. Berger superalgebras are defined and their examples are given.
