  We give an overview of differential cohomology from a modern,
homotopy-theoretic perspective in terms of sheaves on manifolds. Although
modern techniques are used, we base our discussion in the classical precursors
to this modern approach, such as Chern-Weil theory and differential characters,
and include the necessary background to increase accessibility. Special
treatment is given to differential characteristic classes, including a
differential lift of the first Pontryagin class. Multiple applications,
including to configuration spaces, invertible field theories, and conformal
immersions, are also discussed. This book is based on talks given at MIT's
Juvitop seminar jointly with UT Austin in the Fall of 2019.
