The dispersion of heavy inertial particles in statistically stationary stably stratified turbulence is studied by means of direct numerical simulations. The following issues have been addressed: What distinguishes dispersion in such stratified flows from dispersion processes in statistically stationary homogeneous isotropic turbulence? How is the dispersion process affected by the Stokes number of the inertial particles (0.1¿St=tp/tK ¿10, with tp the particle response time and tK the Kolmogorov time)? What is the interplay between buoyancy and the Stokes number? And what is the effect, if any, of particle settling, nonlinear drag, and lift forces (particularly relevant for stratified turbulence with its vertical shear layers) on particle dispersion? The long-time dispersion in isotropic turbulence is found to be maximum around St=1, in agreement with the observation of preferential concentration for St˜1. In stably stratified turbulence such a maximum in the dispersion is only found for the horizontal direction. The horizontal and vertical dispersions in stably stratified turbulence show different behaviors due to the anisotropy of the flow, and in particular, vertical dispersion is strongly affected by the inertia of the particles. With increasing St the classical plateau found for vertical fluid particle dispersion becomes less pronounced and it even vanishes for Stokes numbers of O(10) and higher. Furthermore, the long-time vertical dispersion increases with increasing St. The effects of gravity, nonlinear drag, and lift forces have been considered in more detail. It turned out that the settling enhancement of inertial particles, as observed in isotropic turbulence, is suppressed by stratification and by nonlinear drag effects. Moreover, nonlinear drag only affects the dispersion in the vertical direction in stably stratified turbulence. Finally, it is found that lift forces can safely be neglected for dispersion studies under the current parameter settings.