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 sub(f) approximately -29.3 to -30.2) spanning a redshift range of z approximately 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 unk 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 super(+) sub(-) super(0) sub(0) super(.) sub(.) super(0) sub(0) super(9) sub(8) and constrain any neutral intrinsic absorbing material to have a mean column density of N sub(H) unk 2 x 10 super(21) cm super(-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 alpha sub(ox) = -1.80 plus or minus 0.02, as well as no significant evolution of alpha sub(ox) with redshift. We also comment on 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.