We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to the Boltzmann equation exists. This is done in the Euclidean setting in any dimension $d\geq 2$, and in the periodic setting in dimensions $d\in\{2,3\}$. As a corollary, we also derive the corresponding fluid equations from the the hard sphere dynamics. This executes the original program, proposed in Hilbert's Sixth Problem in 1900, pertaining to the derivation of hydrodynamic equations from colliding particle systems, via the Boltzmann equation as the intermediate step.