  We provide a rigorous derivation of the Boltzmann equation as the mesoscopic
limit of systems of hard spheres, or Newtonian particles interacting via a
short-range potential, as the number of particles $N$ goes to infinity and the
characteristic length of interaction $\e$ simultaneously goes to $0,$ in the
Boltzmann-Grad scaling $N \e^{d-1} \equiv 1.$
  The time of validity of the convergence is a fraction of the average time of
first collision, due to a limitation of the time on which one can prove uniform
estimates for the BBGKY and Boltzmann hierarchies.
  Our proof relies on the fundamental ideas of Lanford, and the important
contributions of King, Cercignani, Illner and Pulvirenti, and Cercignani,
Gerasimenko and Petrina. The main novelty here is the detailed study of
pathological trajectories involving recollisions, which proves the term-by-term
convergence for the correlation series expansion.
