Measurements of dust coupled to the Jovian magnetosphere have been obtained with the dust detector on board the Galileo spacecraft. We report on data obtained during the first four orbits about Jupiter that had flybys of the Galilean satellites: Ganymede (orbits 1 and 2), Callisto (orbit 3), and Europa (orbit 4). The most prominent features observed are highly time variable dust streams recorded throughout the Jovian system. The impact rate varied by up to 2 orders of magnitude with a 5 and 10 hour periodicity, which shows a correlation with Galileo's position relative to the Jovian magnetic field. Around 20 RJ (Jupiter radius, RJ=71, 492 km) in bound a dip in the impact rate has been found consistently. At the same times, reversals by 180¿ in impact direction occurred. This behavior can be qualitatively explained by strong coupling of nanometer-sized dust to the Jovian magnetic field. At times of satellite flybys, enhanced rates of dust impacts have been observed, which suggests that all Galilean satellites are sources of ejecta particles. Inside about 20 RJ impacts of micrometer-sized particles have been recorded that could be particles on bound orbits about Jupiter. ¿ 1998 American Geophysical Union