  We consider differential operators on a supermanifold of dimension $1|1$. We
define non-degenerate operators as those with an invertible top coefficient in
the expansion in the "superderivative" $D$ (which is the square root of the
shift generator, the partial derivative in an even variable, with the help of
an odd indeterminate). They are remarkably similar to ordinary differential
operators. We show that every non-degenerate operator can be written in terms
of `super Wronskians' (which are certain Berezinians). We apply this to Darboux
transformations (DTs), proving that every DT of an arbitrary non-degenerate
operator is the composition of elementary first order transformations. Hence
every DT corresponds to an invariant subspace of the source operator and, upon
a choice of basis in this subspace, is expressed by a super-Wronskian formula.
We consider also dressing transformations, i.e., the effect of a DT on the
coefficients of the non-degenerate operator. We calculate these transformations
in examples and make some general statements.
