We use a Monte Carlo model to simulate impact histories of possible Titans, Callistos, and Ganymedes. Comets create or erode satellite atmospheres, depending on their mass and velocity distributions: faster and bigger comets remove atmophiles; slower or smaller comets supply them. Mass distributions and the minimum total mass of comets passing through the Saturn system were derived from the crater records of Rhea and Iapetus. These were then scaled to give a minimum impact history for Titan. From this cometary population, of 1000 initially airless Titans, 16% acquired atmospheres larger than Titan's present atmosphere (9¿1021 g), and more than half accumulated atmospheres larger than 1021 g. In contrast to the work of Zahnle et al. (1992), we find that, in most trials, Callisto acquires comet-based atmospheres. Atmospheres acquired by Callisto and, especially, Ganymede are sensitive to assumptions regarding energy partitioning into the ejecta plume. If we assume that only the normal velocity component heats the plume, the majority of Ganymedes and half of the Callistos accreted atmospheres smaller than 1020 g. If all the impactor's velocity heats the plume, Callisto's most likely atmosphere is 1017 g and Ganymede's is negligible. The true cometary flux was most likely larger than that derived crater records, which raises the probability that Titan, Ganymede, and Callisto acquired substantial atmospheres. However, other loss processes (e.g., sputtering by ions swept up by the planetary magnetic field, solar UV photolysis of hydrocarbons) are potentially capable of eliminating small atmospheres over the age of the solar system. The dark material on Callisto's surface may be a remnant of an earlier, now vanished atmosphere. ¿ American Geophysical Union 1995