  Deligne cohomology can be viewed as a differential refinement of integral
cohomology, hence captures both topological and geometric information. On the
other hand, it can be viewed as the simplest nontrivial version of a
differential cohomology theory. While more involved differential cohomology
theories have been explicitly twisted, the same has not been done to Deligne
cohomology, although existence is known at a general abstract level. We work
out what it means to twist Deligne cohomology, by taking degree one twists of
both integral cohomology and de Rham cohomology. We present the main properties
of the new theory and illustrate its use with examples and applications. Given
how versatile Deligne cohomology has proven to be, we believe that this
explicit and utilizable treatment of its twisted version will be useful.
