The endeavor to understand how fluid dynamical equations can be derived from kinetic theory goes back to the founding works of Maxwell and Boltzmann. Most of these derivations are well understood at several formal levels by now, and yet their full mathematical justifications are still missing. This talk will introduce this general problem and describe recent works in which the Stokes and acoustic limits are globally establish for the classical Boltzmann equation considered over any periodic spatial domain of dimension two or more.