Utilizing 21 new Chandra observations as well as archival Chandra, ROSAT, and XMM-Newton data, we study the X-ray properties of a representative sample of 59 of the most optically luminous quasars in the Universe (M_i~~-29.3 to -30.2) spanning a redshift range of z~~1.5-4.5. Our full sample consists of 32 quasars from the Sloan Digital Sky Survey (SDSS) Data Release 3 (DR3) quasar catalog, two additional objects in the DR3 area that were missed by the SDSS selection criteria, and 25 comparably luminous quasars at z>~4. This is the largest X-ray study of such luminous quasars to date. By jointly fitting the X-ray spectra of our sample quasars, excluding radio-loud and broad absorption line (BAL) objects, we find a mean X-ray power-law photon index of Gamma=1.92^{+0.09}_{-0.08} and constrain any neutral intrinsic absorbing material to have a mean column density of N_H<~2x10^{21} cm^{-2}. We find, consistent with other studies, that Gamma does not change with redshift, and we constrain the amount of allowed Gamma evolution for the most-luminous quasars. Our sample, excluding radio-loud and BAL quasars, has a mean X-ray-to-optical spectral slope of a_ox=-1.80+/-0.02, as well as no significant evolution of a_ox with redshift. We also comment upon the X-ray properties of a number of notable quasars, including an X-ray weak quasar with several strong narrow absorption-line systems, a mildly radio-loud BAL quasar, and a well-studied gravitationally lensed quasar. Comment: 18 pages (emulateapj), 11 figures. Accepted for publication in The Astrophysical Journal