We report the results of an almost complete survey of the X-ray properties of Hickson's compact galaxy groups with the ROSAT PSPC. Diffuse X-ray emission is detected from 22 groups. We infer that hot intragroup gas is present in greater than or similar to 75 per cent of these systems, and derive their X-ray luminosity function. Earlier reports that only spiral-poor systems exhibit diffuse X-ray emission are found Co be incorrect. Strong correlations are found between the X-ray luminosity and both the gas temperature and the velocity dispersion of the group galaxies. We argue that these properties provide strong evidence that most of these groups are genuinely compact configurations, rather than line-of-sight superpositions. Comparison with the X-ray properties of galaxy clusters indicates a significant steepening of the L-T relation below T similar to 1 KeV, which may result from the action of galaxy winds.