Aims. Hyper-Luminous Infrared Galaxies (HLIRGs) are the most luminous persistent objects in the Universe. They exhibit extremely high star formation rates, and most of them seem to harbor an Active Galactic Nucleus (AGN). They are unique laboratories to investigate ultimate star formation, and its connection to super-massive black hole growth. X-ray studies of HLIRGs have the potential to unravel the AGN contribution to the bolometric output from these bright objects. Methods. We have selected a sample of 14 HLIRGs observed by XMM-Newton (type 1, type 2 AGN and starburst), 5 of which are candidates to be Compton-thick objects. This is the first time that a systematic study of this type of objects has been carried out in the X-ray spectral band. Their X-ray spectral properties have been correlated with their infrared luminosities, estimated by IRAS, ISO and sub-millimeter observations. Results. The X-ray spectra of HLIRGs present heterogeneous properties. All our X-ray detected HLIRGs (10) have AGN-dominated X-ray spectra. The hard X-ray luminosity of 8 of them is consistent with a pure AGN contribution, while in the remaining 2 sources, both an AGN and a starburst seem to contribute to the overall emission. We found soft excess emission in 5 sources. In one of them it is consistent with a pure starburst origin, while in the other 4 sources it is consistent with an AGN origin. The observed X-ray emission is systematically below that expected for a standard local QSO of the same IR luminosity, suggesting the possible presence of absorption in type 2 objects and/or a departure from a standard spectral energy distribution of QSO. The X-ray-to-IR-luminosity ratio is constant with redshift, indicating similar evolutions for the AGN and starburst component, and that their respective power sources could be physically related.