Kinetic theory is a legacy of Boltzmann and Maxwell, and is one of the deepest scientific ideas ever introduced. The question of establishing its mathematical foundations was raised soon after Boltzmann introduced his physical theory, and proved to be a highly interesting and challenging direction of research over the past 120 years. In its classical particle setting, this was part of Hilbert’s sixth problem, which asked for the derivation of the equations of fluid mechanics—such as the Euler and Navier-Stokes equations—from first principles, by rigorously justifying Boltzmann’s kinetic theory as an intermediate step. This entails starting from Newton’s laws for a system of N particles and taking successive limits to first obtain Boltzmann’s kinetic equation, and then deriving the equations of fluid mechanics from it. The major landmark in the early literature is the work of Oscar Lanford (1975), who provided the first rigorous derivation of the Boltzmann equation in the so-called Boltzmann-Grad limit, albeit only for short times. The latter restriction does not allow for a passage to the fluid limit as suggested in Hilbert’s sixth problem. In recent joint work with Yu Deng (University of Chicago) and Xiao Ma (University of Michigan), we extend Lanford’s theorem to long times—specifically, for as long as the solution of the Boltzmann equation exists. This allows us to carry out the fluid limit, and derive the fluid equations in the Boltzmann–Grad limit. The underlying strategy builds on earlier joint work with Yu Deng that resolved the parallel problem in wave kinetic theory (also known as wave turbulence theory), in which colliding particles are replaced by nonlinear waves. In this talk, we will review this progress and discuss several future directions that remain unsolved, and still lie under the broad umbrella of Hilbert’s sixth problem.