Tenuous dust clouds of Jupiter's Galilean moons Io, Europa, Ganymede and Callisto have been detected with the in-situ dust detector on board the Galileo spacecraft. The majority of the dust particles have been sensed at altitudes below five radii of these lunar-sized satellites. We identify the particles in the dust clouds surrounding the moons by their impact direction, impact velocity, and mass distribution. Average particle sizes are 0.5 to $\rm 1 \mu m$, just above the detector threshold, indicating a size distribution with decreasing numbers towards bigger particles. Our results imply that the particles have been kicked up by hypervelocity impacts of micrometeoroids onto the satellites' surfaces. The measured radial dust density profiles are consistent with predictions by dynamical modeling for satellite ejecta produced by interplanetary impactors (Krivov et al., PSS, 2003, 51, 251--269), assuming yield, mass and velocity distributions of the ejecta from laboratory measurements. The dust clouds of the three outer Galilean moons have very similar properties and are in good agreement with the model predictions for solid ice-silicate surfaces. The dust density in the vicinity of Io, however, is more than an order of magnitude lower than expected from theory. This may be due to a softer, fluffier surface of Io (volcanic deposits) as compared to the other moons. The log-log slope of the dust number density in the clouds vs. distance from the satellite center ranges between --1.6 and --2.8. Appreciable variations of number densities obtained from individual flybys with varying geometry, especially at Callisto, might be indicative of leading-trailing asymmetries of the clouds due to the motion of the moons with respect to the field of impactors.