The Boltzmann equation is one of the central equations in statistical mechanics and models the evolution of a gas through particle interactions. In recent years, groundbreaking work by Imbert and Silvestre has led to a conditional regularity theory for periodic solutions of the Boltzmann equation. They established that any possible singularity of a periodic solution to the Boltzmann equation must be visible macroscopically. A major open challenge is whether such a theory can be extended to bounded domains with physically relevant boundary conditions. In this talk, I will first give an accessible overview of the conditional regularity program by Imbert and Silvestre, highlighting its main ideas and implications. As a first step toward understanding the boundary case, I will then discuss the smooth- ness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection and in-flow condition. While the interior regularity of such equations is well understood, their behavior near the boundary has remained open, even in the simplest case of Kolmogorov’s equation. Finally, I will report on recent joint works with Xavier Ros-Oton and Kyeongbae Kim, in which we establish sharp boundary regularity results for this class of equations.