**Abstract** : Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re " 1 and horizontal Froude number Fh Gt; 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter R = ReFh2. When R " 1, viscous forces are nimportant and lv scales as lv ~ U/N (U is a characteristic horizontal velocity and N is the Brunt - Väis¨alä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When R " 1, vertical viscous shearing is important so that lv ~ lh/Re1/2 (lh is a characteristic horizontal length scale). The parameter R is further shown to be related to the buoyancy Reynolds number and proportional to (lO/?) 4/3, where lO is the Ozmidov length scale and ? the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when R " 1: the scales larger than lO are strongly influenced by the stratification while those between lO and ? are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being R. The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of R but they tend to be smooth for R > 1, while for R > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for R > 1 but tends to isotropy as R increases above unity. When R > 1, the horizontal and vertical energy spectra are very steep while, when R > 1, the horizontal spectra of kinetic and potential energy exhibit an pproximate kh-5/3-power-law range and a clear forward energy cascade is observed. © 2007 Cambridge University Press.