References for differential cohomology and differential characters I am interested in learning differential cohomology and differential characters, and am currently studying these lecture notes 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 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. 
 A: The following book has recently appeared:

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*Araminta Amabel, Arun Debray, Peter J. Haine (eds.), Differential Cohomology: Categories, Characteristic Classes, and Connections. Based on Fall 2019 talks at MIT's Juvitop seminar by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. (arXiv:2109.12250)

A: 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
