In this minicourse, we will describe the main ideas of our recent joint work with Yu Deng (Chicago) and Xiao Ma (Michigan) aimed at giving the rigorous derivation of the Boltzmann equation from Newton's laws on colliding particle systems, for arbitrarily long times. This leads to the full execution of the so-called "Hilbert’s Program", proposed in Hilbert's sixth problem from 1900, and aimed at deriving the fundamental equations of fluid dynamics from Newton’s laws, with Boltzmann’s equation as an intermediate step. The result follows parallel progress by Yu Deng and myself in the wave setting, where colliding particles are replaced by interacting waves, which we will briefly discuss as well.