We present the results of two XMM-Newton observations of Jupiter carried out in 2003 for 100 and 250 ks (or 3 and 7 planet rotations) respectively. X-ray images from the EPIC CCD cameras show prominent emission from the auroral regions in the 0.2 - 2.0 keV band: the spectra are well modelled by a combination of emission lines, including most prominently those of highly ionised oxygen (OVII and OVIII). In addition, and for the first time, XMM-Newton reveals the presence in both aurorae of a higher energy component (3 - 7 keV) which is well described by an electron bremsstrahlung spectrum. This component is found to be variable in flux and spectral shape during the Nov. 2003 observation, which corresponded to an extended period of intense solar activity. Emission from the equatorial regions of Jupiter's disk is also observed, with a spectrum consistent with that of solar X-rays scattered in the planet's upper atmosphere. Jupiter's X-rays are spectrally resolved with the RGS which clearly separates the prominent OVII contribution of the aurorae from the OVIII, FeXVII and MgXI lines, originating in the low-latitude disk regions of the planet.