The statistics of Lagrangian particles transported by a three-dimensional fully developed turbulent flow is investigated by means of high-resolution direct numerical simulations. The analysis of single trajectories reveals the existence of strong trapping events vortices at the Kolmogorov scale which contaminates inertial range statistics up to 10 t¿. For larger time separations, we find that Lagrangian structure functions display intermittency in agreement with the prediction of the multifractal model of turbulence. The study of two-particle dispersion shows that the probability density function of pair separation is very close to the original prediction of Richardson of 1926. Nevertheless, moments of relative dispersion are strongly affected by finite Reynolds effects, thus limiting the possibility to measure numerical prefactors, such as the Richardson constant g. We show how, by using an exit time statistics, it is possible to have a precise estimation of g which is consistent with recent laboratory measurements.