  We introduce a new method for ``twisting'' relative equivalences of derived
categories of sheaves on two spaces over the same base. The first aspect of
this is that the derived categories of sheaves on the spaces are twisted. They
become derived categories of sheaves on gerbes living over spaces that are
locally (on the base) isomorphic to the original spaces. Secondly, this is done
in a compatible way so that the equivalence is maintained. We apply this method
by proving the conjectures of Donagi and Pantev on dualities between gerbes on
genus-one fibrations and comment on other applications to families of higher
genus curves. We also include a related conjecture in Mirror Symmetry. This
represents a modified version of my May 2006 thesis for the University of
Pennsylvania. The only addition is the subsection titled 'Alternative Method'.
