We present two XMM-Newton observations of Jupiter, which were carried out in April and November 2003 for 110 and 250 ks (or 3 and 7 planet rotations) respectively. X-ray images taken with XMM-Newton EPIC CCD cameras show prominent emission, essentially all confined to the 0.2 - 2.0 keV band, from Jupiter's auroral spots; their spectra can be modelled with a combination of unresolved emission lines, including most prominently those of highly ionised oxygen (OVII and OVIII). Emission from the equatorial regions of the planet's disk is also observed. Its spectrum is consistent with that of solar X-rays scattered in the planet's upper atmosphere. More remarkably, we find that in November 2003 a large solar X-ray flare, taking place on the Sun's Jupiter-facing side, is associated with a corresponding feature in the Jovian X-ray lightcurve of the equatorial regions. Jupiter's X-ray emissions are spectrally resolved with XMM-Newton Reflection Grating Spectrometer (RGS). The high energy resolution provided by RGS allows us to clearly separate the OVII and OVIII lines in the spectra, and to identify most of the auroral emission with the lower ionisation line. The North auroral spot emission is deeply modulated at the planet's rotation period. Moreover, the X-ray emission from Jupiter's disk displays prominent line contribution due to FeXVII, in addition to an OVIII component. Our temporal and spectral findings suggest that the non-auroral X-ray emission from Jupiter is directly controlled by the Sun. On the other hand, the spectral results presented here support the hypothesis that Jupiter's auroral emissions originate from the capture and acceleration of solar wind ions in the planet's magnetosphere, followed by X-ray production by charge exchange.