The eROSITA Final Equatorial-Depth Survey has been carried out during the PV phase of the SRG/eROSITA telescope and completed in November 2019. This survey is designed to provide the first eROSITA-selected sample of galaxy clusters and groups and to test the predictions for the all-sky survey in the context of cosmological studies with clusters. In the 140 deg$^2$ area covered by eFEDS, 542 candidate clusters and groups are detected as extended X-ray sources, down to a flux of $sim10^{-14} $erg/s/cm$^2$ in the soft band (0.5-2 keV) within 1. In this work, we provide the catalog of candidate galaxy clusters and groups in eFEDS. We perform imaging and spectral analysis on the eFEDS clusters with eROSITA X-ray data, and study the properties of the sample. The clusters are distributed in the redshift range [0.01, 1.3], with the median redshift at 0.35. We obtain the ICM temperature measurement with $>2sigma$ c.l. for $sim$1/5 (102/542) of the sample. The average temperature of these clusters is $sim$2 keV. Radial profiles of flux, luminosity, electron density, and gas mass are measured from the precise modeling of the imaging data. The selection function, the purity and completeness of the catalog are examined and discussed in detail. The contamination fraction is $sim1/5$ in this sample, dominated by misidentified point sources. The X-ray Luminosity Function of the clusters agrees well with the results obtained from other recent X-ray surveys. We also find 19 supercluster candidates in eFEDS, most of which are located at redshifts between 0.1 and 0.5. The eFEDS cluster and group catalog provides a benchmark proof-of-concept for the eROSITA All-Sky Survey extended source detection and characterization. We confirm the excellent performance of eROSITA for cluster science and expect no significant deviations from our pre-launch expectations for the final All-Sky Survey.
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