We present the third edition of the Sloan Digital Sky Survey (SDSS) Quasar Catalog. The catalog consists of the 46,420 objects in the SDSS Third Data Release that have luminosities larger than Mi = -22 (in a cosmology with H0 = 70 km s-1 Mpc-1, ΩM = 0.3, and ΩΛ = 0.7), have at least one emission line with FWHM larger than 1000 km s-1 or are unambiguously broad absorption line quasars, are fainter than i = 15.0, and have highly reliable redshifts. The area covered by the catalog is ≈4188 deg2. The quasar redshifts range from 0.08 to 5.41, with a median value of 1.47; the high-redshift sample includes 520 quasars at redshifts greater than 4, of which 17 are at redshifts greater than 5. For each object the catalog presents positions accurate to better than 0farcs2 rms per coordinate, five-band (ugriz) CCD-based photometry with typical accuracy of 0.03 mag, and information on the morphology and selection method. The catalog also contains radio, near-infrared, and X-ray emission properties of the quasars, when available, from other large-area surveys. The calibrated digital spectra cover the wavelength region 3800–9200 Å at a spectral resolution of ≃2000; the spectra can be retrieved from the public database using the information provided in the catalog. A total of 44,221 objects in the catalog were discovered by the SDSS; 28,400 of the SDSS discoveries are reported here for the first time.