  The main goal of the present paper is the construction of twisted generalized
differential cohomology theories and the comprehensive statement of its basic
functorial properties. Technically it combines the homotopy theoretic approach
to (untwisted) generalized differential cohomology developed by Hopkins-Singer
and later by the first author and D. Gepner with the oo-categorical treatement
of twisted cohomology by Ando-Blumberg-Gepner.
  We introduce the notion of a differential twist for a given generalized
cohomology theory and construct twisted differential cohomology groups (resp.
spectra). The main technical results of the paper are existence and uniqueness
statements for differential twists. These results will be applied in a variety
of examples, including K-theory, topological modular forms and other cohomology
theories.
