Abstract: The goal of this lecture is to show how the brownian motion can be derived rigorously from a deterministic system of hard spheres in the limit where the number of particles $N$ tends to infinity, and their diameter simultaneously converges to 0. As suggested by Hilbert in his sixth problem, we will use the linear Boltzmann equation as an intermediate level of description for the dynamics of one tagged particle. We will discuss especially the origine of irreversibility, which is a fundamental feature of both the brownian motion and the Boltzmann equation having no counterpart at the microscopic level.