Interplanetary and interstellar dust particles stream through the Jovian magnetosphere where their orbits are perturbed by the Lorentz force. Energy and angular momentum exchange with the magnetosphere can lead to capture of some submicron dust particles into dynamically stable orbits around Jupiter. The captured grains can be on prograde or retrograde orbits, and their final orbits are usually in the semimajor axis range of 2 to 20 Jupiter radii, with eccentricities ~0.1--0.8 and inclinations ≲20¿. Captured grains have radii between about 0.5 and 1.5 μm. These captured dust particles form a diffuse band of material around Jupiter, outside the main Jovian ring, with a peak normal optical depth of &tgr;≲10-9. While this is too low for optical detection, the Galileo dust detector has detected particles on prograde and retrograde orbits around Jupiter whose sizes, orbital properties, and number density are consistent with captured interplanetary and interstellar dust. ¿ 1998 American Geophysical Union