  We present a probabilistic proof of the mean field limit and propagation of
chaos $N$-particle systems in three dimensions with positive (Coulomb) or
negative (Newton) $1/r$ potentials scaling like $1/N$ and an $N$-dependent
cut-off which scales like $N^{-1/3+ \epsilon}$. In particular, for typical
initial data, we show convergence of the empirical distributions to solutions
of the Vlasov-Poisson system with either repulsive electrical or attractive
gravitational interactions.
