I am interested in learning differential cohomology and differential characters, and am currently studying these [lecture notes](https://arxiv.org/abs/1208.3961) on the subject. I sometimes feel it would be great if I could keep some more good references besides the lecture notes, as it could greatly help me speed up when I get stuck (which is happening often with me). Could someone please advise some alternate references which might supplement my study ?


I am familiar with elementary algebraic topology (singular homology & cohomology theories, basic homotopy theory) and differential geometry (connections, curvature, de-Rham cohomology, Chern-Weil theory).


Thanks ! 


Note : I earlier [asked](https://math.stackexchange.com/questions/1964374/references-for-differential-cohomology-and-secondary-characteristic-classes) this question on m.se and offered a bounty but I did not get any answers there, I am taking the liberty of asking here. 



References for differential cohomology and differential characters
Bunke's notes are indeed a great source for this material! However, in order to get to the main definitions and properties, he does breeze through a lot of the fundamental prerequisites. I will briefly add to the great list Omar has provided above.


For a classical, non sheaf theoretic approach: Jeff Cheeger, James Simons, *Differential characters and geometric invariants*. <http://numr.wdfiles.com/local--files/differential-cohomology/cheeger-simons.pdf>


For a great text which models differential cohomology via Deligne cohomology (and discusses the relationship with $U(1)$-gerbes in low degrees): Jean-Luc Brylinski, *loop spaces characteristic classes and geometric quantization*. 


If you have a strong background in category theory (in particular categories of sheaves), I would strongly recommend the higher stack model of differential cohomology (similar to Brylinski's gerbes, but generalized to higher dimensions) which is discussed in: Urs Schreiber, *Differential cohomology in a cohesive infinity-topos* <https://arxiv.org/pdf/1310.7930.pdf>


For the original definition in terms of differnetial function spectra, you can see: Mike Hopkins, Isadore Singer, *Quadratic Functions in Geometry, Topology,and M-Theory* <https://arxiv.org/abs/math/0211216>


If you want to see a brief summary of Bunke's model for differential cohomology theories (the papers that Omar mentioned are far more comprehensive and thorough, but just to add a bit more to the list), there is also a short discussion in this pre-print of mine with Hisham Sati: *Spectral sequences in smooth generalized cohomology* <https://arxiv.org/pdf/1605.03444.pdf>


