Remote observations with the Chandra X-ray Observatory and the XMM-Newton Observatory have shown that the jovian system is a source of X-rays with a rich and complicated structure. The planet's polar auroral zones and its disk are both powerful sources of X-ray emission. Chandra observations revealed X-ray emission from the Io plasma torus and from the Galilean moons Io, Europa, and possibly Ganymede. The emission from the moons is due to bombardment of their surfaces by highly energetic magnetospheric protons, and oxygen and sulfur ions. These ions excite atoms in their surfaces leading to fluorescent X-ray emission lines. These lines are produced against an intense background continuum, including bremsstrahlung radiation from surface interactions of primary magnetospheric and secondary electrons. Although the X-ray emission from the Galilean moons is faint when observed from Earth orbit, an imaging X-ray spectrometer in orbit around one or more of these moons, operating from 200 eV to 8 keV with 150 eV energy resolution, would provide a detailed mapping of the elemental composition in their surfaces. Surface resolution of 40 m for small features could be achieved in a 100-km orbit around one moon while also remotely imaging surfaces of other moons and Jupiter's upper atmosphere at maximum regional resolutions of hundreds of kilometers. Due to its relatively more benign magnetospheric radiation environment, its intrinsic interest as the largest moon in the Solar System, and its mini-magnetosphere, Ganymede would be the ideal orbital location for long-term observational studies of the jovian system. Here we describe the physical processes leading to X-ray emission from the surfaces of Jupiter's moons and the properties required for the technique of imaging X-ray spectroscopy to map the elemental composition of their surfaces, as well as studies of the X-ray emission from the planet's aurora and disk and from the Io plasma torus.