  We analyse in detail the language of partially non-abelian Deligne cohomology
and of twisted differential K-theory, in order to describe the geometry of type
II superstring backgrounds with D-branes. This description will also provide
the opportunity to show some mathematical results of independent interest. In
particular, we begin classifying the possible gauge theories on a D-brane or on
a stack of D-branes using the intrinsic tool of long exact sequences.
Afterwards, we recall how to construct two relevant models of differential
twisted K-theory, paying particular attention to the dependence on the twisting
cocycle within its cohomology class. In this way we will be able to define
twisted K-homology and twisted Cheeger-Simons K-characters in the category of
simply-connected manifolds, eliminating any unnatural dependence on the
cocycle. The ambiguity left for non simply-connected manifolds will naturally
correspond to the ambiguity in the gauge theory, following the previous
classification. This picture will allow for a complete characterization of
D-brane world-volumes, the Wess-Zumino action and topological D-brane charges
within the K-theoretical framework, that can be compared step by step to the
old cohomological classification. This has already been done for backgrounds
with vanishing B-field; here we remove this hypothesis.
