Relative dispersion in fully developed turbulence is investigated by means of direct numerical simulations. Lagrangian statistics is found to be compatible with Richardson description although small systematic deviations are found. The value of the Richardson constant is estimated as C2 approximately equal to 0.55, in a close agreement with recent experimental findings [S. Ott and J. Mann, J. Fluid Mech. 422, 207 (2000)]. By means of exit-time statistics it is shown that the deviations from Richardson's law are a consequence of Eulerian intermittency. The measured Lagrangian scaling exponents require a set of Eulerian structure function exponents zeta(p) which are remarkably close to standard ones known for fully developed turbulence.