Cross-tropopause transport of O-3 is a significant factor in the tropospheric budget and distribution of O-3. Nevertheless, the distribution in the troposphere of O-3 that originates from the stratosphere is uncertain. We study this with a chemistry - general circulation model with relatively high spatial and temporal resolution. The model simulates background tropospheric CH4-CO-NOx-HOx photochemistry, and includes a tracer for stratospheric O-3. Since this tracer is not photochemically produced in the troposphere but only destroyed, comparing its budget and distribution with that of total tropospheric O-3 yields an estimate of the contribution of the stratospheric O-3 source in the troposphere. Model results suggest that transport from the stratosphere and net photochemical formation in the troposphere, considering present-day emissions, are of comparable magnitude. The model predicts efficient transport of upper tropospheric O-3-rich air to the surface by large-scale subsidence in the subtropics and by synoptic disturbances in the NH middle and high latitudes. O-3 from the stratosphere contributes significantly to surface Of in winter and spring when the photochemical lifetime of O-3 is relatively long. In summer and in the tropics, little O-3 from the stratosphere reaches the surface due to strong photochemical destruction, so that surface O-3 is largely determined by photochemical production. Photochemically produced O-3 maximizes in the free troposphere where the O-3 surface warming efficiency is higher compared to the boundary layer.