XMM-Newton observations of 10 ULIRGs are reported. The aim is to investigate in hard X-rays a complete ULIRG sample selected from the bright IRAS 60$mu$m catalogue. All sources are detected in X-rays, 5 of which for the first time. These observations confirm that ULIRGs are intrinsically faint X-rays sources, their observed X-ray luminosities being typically L(2-10 keV)<1E42-43 erg/s, whereas their bolometric luminosities are L>1E45 erg/s. In all sources we find evidence for thermal emission from hot plasma with kT~0.7keV, dominating the X-ray spectra below 1keV, and likely associated with a nuclear or circumnuclear starburst. This thermal emission appears uncorrelated with the FIR luminosity, suggesting that,in addition to the ongoing rate of star formation, other parameters may also affect it. The soft X-ray emission appears to be extended on a scale of ~30kpc for Mkn231 and IRAS19254-7245, possible evidence of galactic superwinds. In these 2 sources, in IRAS20551-4250 and IRAS23128-5919 we find evidence for the presence of hidden AGNs, while a minor AGN contribution may be suspected also in IRAS20100-4156. In particular, we have detected a strong Fe line at 6.4keV in the spectrum of IRAS19254-7245 and a weaker one in Mkn231, suggestive of deeply buried AGNs. For the other sources, the X-ray luminosities and spectral shapes are consistent with hot thermal plasma and X-ray binary emissions of mainly starburst origin. We find that the 2-10keV luminosities in these sources, most likely due to high-mass X-ray binaries, are correlated with L_FIR: both luminosities are good indicators of the current global SFR in the galaxy. The composite nature of ULIRGs is then confirmed, with hints for a predominance of the starburst over the AGN phenomenon in these objects.