We present the Bright SHARC (Serendipitous High-Redshift Archival ROSAT Cluster) Survey, which is an objective search for serendipitously detected extended X-ray sources in 460 deep ROSAT PSPC pointings. The Bright SHARC Survey covers an area of 178.6 sq.deg and has yielded 374 extended sources. We discuss the X-ray data reduction, the candidate selection and present results from our on-going optical follow-up campaign. The optical follow-up concentrates on the brightest 94 of the 374 extended sources and is now 97% complete. We have identified thirty-seven clusters of galaxies, for which we present redshifts and luminosities. The clusters span a redshift range of 0.0696<z<0.83 and a luminosity range of 0.065<Lx<8.3e44 erg/s [0.5-2.0 keV] (assuming Ho = 50 km/s/Mpc and qo=0.5). Twelve of the clusters have redshifts greater than z=0.3, eight of which are at luminosities brighter than Lx=3e44 erg/s. Seventeen of the 37 optically confirmed Bright SHARC clusters have not been listed in any previously published catalog. We also report the discovery of three candidate ``fossil groups of the kind proposed by Ponman et al. (1994).
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