The Boltzmann and Landau equations are two fundamental models in kinetic theory. They are nonlocal and nonlinear equations for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk, I will discuss two more tractable and related questions: (1) Local well-posedness for the Boltzmann equation and (2) Schauder estimates and their application to uniqueness of solutions to the Landau equation. At the end of the talk, I will discuss some open problems and future work. This is based on joint work with Christopher Henderson.