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 deg ^ { 2 } 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 < L _ { x } < 8.3 \times 10 ^ { 44 } erg s ^ { -1 } [ 0.5-2.0 keV ] ( assuming H _ { 0 } = 50 km s ^ { -1 } Mpc ^ { -1 } and q _ { 0 } = 0.5 ) . Twelve of the clusters have redshifts greater than z =0.3 , eight of which are at luminosities brighter than L _ { x } = 3 \times 10 ^ { 44 } erg s ^ { -1 } . 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 ( 80 ) .