While the terrestrial aurorae are known to be driven primarily by the interaction of the Earth's magnetosphere with the solar wind, there is considerable evidence that auroral emissions on Jupiter and Saturn are driven primarily by internal processes, with the main energy source being the planets' rapid rotation. Prior observations have suggested there might be some influence of the solar wind on Jupiter's aurorae and indicated that auroral storms on Saturn can occur at times of solar wind pressure increases. To investigate in detail the dependence of auroral processes on solar wind conditions, a large campaign of observations of these planets has been undertaken using the Hubble Space Telescope, in association with measurements from planetary spacecraft and solar wind conditions both propagated from 1 AU and measured near each planet. The data indicate a brightening of both the auroral emissions and Saturn kilometric radiation at Saturn close in time to the arrival of solar wind shocks and pressure increases, consistent with a direct physical relationship between Saturnian auroral processes and solar wind conditions. At Jupiter the correlation is less strong, with increases in total auroral power seen near the arrival of solar wind forward shocks but little increase observed near reverse shocks. In addition, auroral dawn storms have been observed when there was little change in solar wind conditions. The data are consistent with some solar wind influence on some Jovian auroral processes, while the auroral activity also varies independently of the solar wind. This extensive data set will serve to constrain theoretical models for the interaction of the solar wind with the magnetospheres of Jupiter and Saturn.