We perform a survey of the X-ray properties of 41 objects from the WISE/SDSS selected Hyper-luminous (WISSH) quasars sample, which includes 86 broad-line quasars with bolometric luminosity L Bol > ∼ 2 × 10 47 erg s −1 shining at z ∼ 2-4. We use both proprietary and archival Chandra and XMM-Newton observations. Twenty-one quasars have sufficient quality data to perform a spectroscopic analysis, while for the remaining sources, X-ray properties are derived through hardness-ratio analysis (apart for six sources which result to be undetected). The bulk (∼ 70%) of the detected WISSH quasars exhibit N H < 5 × 10 22 cm −2 , in agreement with their optical Type 1 AGN classification. All but three quasars show unabsorbed 2-10 keV luminosities L 2−10 ≥ 10 45 erg s −1 . Thanks to their extreme radiative output across the Mid-IR-to-X-ray range, WISSH quasars therefore offer the opportunity to significantly extend and validate the existing relations involving L 2−10 . Specifically, we study the X-ray luminosity as a function of (i) X-ray-to-Optical (X/O) flux ratio, (ii) mid-IR luminosity (L MIR ), (iii) L Bol as well as (iv) α OX versus the 2500Å luminosity. We find that the WISSH quasars show (i) unreported very low X/O (< 0.1) compared to typical AGN values; (ii) L 2−10 /L MIR ratios significantly smaller than those derived for AGN with lower luminosity; (iii) a large X-ray bolometric correction k Bol,X ≈ 100-1000; and (iv) steep -2 > ∼ α OX > ∼ -1.7. These results lead to a scenario in which the X-ray emission of hyper-luminous quasars is relatively weaker compared to lower-luminosity AGN. Models predict that such an X-ray weakness can be relevant for the acceleration of powerful high-ionization emission line-driven winds, commonly detected in the UV spectra of WISSH quasars, which can in turn perturb the X-ray corona and weaken its emission. Accordingly, hyper-luminous quasars represent the ideal laboratory to study the link between the AGN energy output and wind acceleration. Additionally, WISSH quasars exhibit very large SMBH masses (log[M BH /M ] > ∼ 9.5). This enables a more robust modeling of the Γ-M BH relation by increasing the statistics at high masses. We derive a flatter Γ dependence than previously found over the broad range 5 < ∼ log(M BH /M ) < ∼ 11. Finally, we estimate that only 300 ks observation of X-IFU on board Athena will offer a detailed view of the properties of absorption features associated to powerful X-ray SMBH winds for a representative sample of WISSH quasars.