Clusters of galaxies in most of the previous catalogs have redshifts z <= 0.3. Using the photometric redshifts of galaxies from the Sloan Digital Sky Survey Data Release 6 (SDSS DR6), the authors identify 39,716 clusters in the redshift range 0.05 < z < 0.6 with more than eight luminous (M_r <= -21) member galaxies. Cluster redshifts are estimated accurately with an uncertainty of less than 0.022. The contamination rate of member galaxies is found to be roughly 20%, and the completeness of member galaxy detection reaches ~90%. Monte Carlo simulations show that the cluster detection rate is more than 90% for massive (M_200 > 2 x 10^14 M_sun, where M_200 is the total mass within the radius in which the mean mass density is 200 times the critical cosmic mass density) clusters of z <= 0.42. The false detection rate is ~5%. The authors obtain the richness, the summed luminosity, and the gross galaxy number within the determined radius for identified clusters. They are tightly related to the X-ray luminosity and temperature of the clusters. Cluster mass is related to the richness and summed luminosity with M_200 ~ R^(1.90+/-0.04)^ and M_200 ~ L_r^(1.64+/-0.03)^, respectively. In addition, 790 new candidate X-ray clusters are found by cross-identification of these clusters with the source list of the ROSAT X-ray All-Sky Survey. This table was created by the HEASARC in August 2009 based on electronic versions of Tables 1 and 2 from the paper which were obtained from the Astrophysical Journal web site. This is a service provided by NASA HEASARC .