The dispersion of fluid particles in statistically stationary stably stratified turbulence is studied by means of direct numerical simulations. Due to anisotropy of the flow, horizontal and vertical dispersion show different behavior. Single-particle dispersion in horizontal direction is similar to that in isotropic turbulence for short times, but shows a long-time growth rate proportional to t^2.1±0.1, larger than the classical linear diffusion limit. In vertical direction, three successive regimes can be identified: a classical t^2-regime, a plateau that scales as N^−2, and a diffusion limit where dispersion is proportional to t. By forcing the flow and performing long-time simulations, we are able to observe this last regime, which was predicted but not observed before in stratified turbulence. This diffusive regime is caused by molecular diffusion of the active scalar (density). The mean squared separation of particle pairs (relative dispersion) in vertical direction shows two plateaus that are not present in isotropic turbulence. They can be associated with the characteristic layered structure of the flow. In the long-time limit again a linear regime is found as for single-particle dispersion. Pair dispersion in horizontal direction behaves similar to that in isotropic turbulence except for long times. Finally, the study of multiparticle statistics in stably stratified turbulent flows is reported. The evolution of tetrads gives an impression of the shape of particle clouds. It is found that with increasing stratification, the volume of the tetrads decreases, and they become flatter and more elongated.