In this work, a magnetically confined e+ beam was crossed with a gas jet to study ionisation of Ne, Kr and Xe atoms. Ions produced in the overlap of the beams were detected in coincidence with the scattered projectile, so as to discriminate against ionisation events involving positronium (Ps) formation. The single ionisation cross section (σi+) and the ratio of the double to single ionisation cross sections (R2) were measured for each target from near threshold to IkeV. By multiplying the R2 values by the appropriate σi+, double ionisation cross section (σi2+) data were also obtained. Comparisons were made with existing data for He and Ar and for e-, p+ and p- impact, to investigate the effects of the projectile mass and charge and the target atomic number (Z). The ratio between σi+ maxima for e+ and e was generally found to be lower for targets with higher Z, whereas no such effect was observed for p+ and p-. This might be due to the deflection of the light projectiles by the target nucleus. In contrast to the charge dependence exhibited by low Z targets, R2(e+) and R2(e-) for Xe were found to have merged at high velocities. This is thought to be due to the increasing importance of inner-shell ionisation followed by Auger decay for heavier atoms. An atomic hydrogen source was incorporated into the system for the investigation of e+ impact ionisation of H. The total ionisation cross section (σPs), including contributions from both direct ionisation and Ps formation, was measured by extracting ions randomly. The Ps formation cross section (σPs) of H was then determined from these measurements by subtraction of data. Two recent theoretical calculations, generally considered to be the most reliable thus far, are found to be in agreement with the results for aps over the entire energy range investigated. The present measurements also differentiate between two prior conflicting measurements of aps, thus providing a degree of consensus amongst experimental data.