Discrete subgroups of Lie groups are much studied and appear throughout mathematics. Anosov subgroups form a class which is intermediate between lattices in higher rank semisimple Lie groups and Fuchsian subgroups of SL(2,R) that uniformize Riemann surfaces. After providing the necessary background, I will explain how Anosov representations can arise as monodromies of families of algebraic manifolds and how this phenomenon is related to Hodge theory. I will then describe some uniformization results for "non-classical" variations of Hodge structure and explain some Torelli theorems for certain 1-dimensional families of Calabi-Yau manifolds.