This study uses turbulence data collected over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean Experiment (SHEBA). Turbulent and mean meteorological data collected at five levels on the 20-m tower are analyzed to examine different regimes of the stable boundary layer (SBL). Eleven months of measurements during SHEBA cover a wide range of the stability conditions, from the weakly unstable regime to the very stable stratification. In the weakly stable regime, the downward heat flux increases with increasing stability parameter, z/L. The traditional Monin - Obukhov stability theory (MOST) works well in this regime. The maximum of the downward heat flux defines the stability boundary between the weakly stable and transition regimes in the SBL. In the transition stability regime the downward heat flux, the drag coefficient, and the turbulence intensity decrease rapidly with increasing z/L due to the buoyancy constraints on the vertical transfer. In the very stable regime, z/L > O(1), turbulence is weak and traditional MOST cannot be applied to describe the turbulence. The vertical divergence of the flux cannot be neglected in the very stable regime and often in the transition regime. Usually the very stable regime is associated with light winds and clear skies. Turbulence may be layered and perhaps intermittent even near the surface. The elevated layers may be decoupled. We observed a two layer structure with weak turbulence occupying the near surface layer (usually 2-3 lowest sonic levels) and collapsed turbulence (no turbulence) above (1-2 upper sonic levels). For the very stable regime, the fluxes and variances may be contaminated by internal gravity waves with periods of several minutes. Some cases in the very stable regime are characterized by small but still significant heat flux and negligibly small stress.