Faltings' Theorem says that a hyperbolic curve has finitely many rational points; concretely, a nonsingular two-variable polynomial equation of degree at least 5 has finitely many rational solutions. The Quadratic Chabauty method has in recent years allowed us to provably find sets of rational points on previously inaccessible hyperbolic curves, but it still has limitations. Quadratic Chabauty is based on the more general non-abelian Chabauty method of Minhyong Kim, but this method has rarely been applied outside the "quadratic" case. We discuss a variety of work in progress, some joint with Ishai Dan-Cohen and/or Martin Lüdtke, which provide the tools for applying the method more generally. Our methods rely heavily on Tannakian categories of Galois representations or motives and their p-adic realizations and periods.