Generalized differential cohomology theories, in particular differential
K-theory (often called "smooth K-theory"), are becoming an important tool in
differential geometry and in mathematical physics. In this survey, we describe
the developments of the recent decades in this area. In particular, we discuss
axiomatic characterizations of differential K-theory (and that these uniquely
characterize differential K-theory). We describe several explicit
constructions, based on vector bundles, on families of differential operators,
or using homotopy theory and classifying spaces. We explain the most important
properties, in particular about the multiplicative structure and push-forward
maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer
family index theorem for differential K-theory.Comment: 50 pages, report based in particular on work done sponsored the DFG
SSP "Globale Differentialgeometrie". v2: final version (only typos
corrected), to appear in C. B\"ar et al. (eds.), Global Differential
Geometry, Springer Proceedings in Mathematics 17, Springer-Verlag Berlin
Heidelberg 201
