The eROSITA telescope array aboard the Spektrum Roentgen Gamma (SRG) satellite began surveying the sky in December 2019, with the aim of producing all-sky X-ray source lists and sky maps of an unprecedented depth. Here we present catalogues of both point-like and extended sources using the data acquired in the first six months of survey operations (eRASS1; completed June 2020) over the half sky whose proprietary data rights lie with the German eROSITA Consortium. We describe the observation process, the data analysis pipelines, and the characteristics of the X-ray sources. With nearly 930 000 entries detected in the most sensitive 0.2- 2.3 keV energy range, the eRASS1 main catalogue presented here increases the number of known X-ray sources in the published literature by more than 60%, and provides a comprehensive inventory of all classes of X-ray celestial objects, covering a wide range of physical processes. A smaller catalogue of 5466 sources detected in the less sensitive but harder 2.3- 5 keV band is the result of the first true imaging survey of the entire sky above 2 keV. We present methods to identify and flag potential spurious sources in the catalogues, which we applied for this work, and we tested and validated the astrometric accuracy via cross-comparison with other X-ray and multi-wavelength catalogues. We show that the number counts of X-ray sources in eRASSl are consistent with those derived over narrower fields by past X-ray surveys of a similar depth, and we explore the number counts variation as a function of the location in the sky. Adopting a uniform all-sky flux limit (at 50% completeness) of F_{05- 2 keV} > 5 × 10^-14 erg s^-1 cm^-2, we estimate that the eROSITA all-sky survey resolves into individual sources about 20% of the cosmic X-ray background in the 1- 2 keV range. The catalogues presented here form part of the first data release (DR1) of the SRG/eROSITA all-sky survey. Beyond the X-ray catalogues, DR1 contains all detected and calibrated event files, source products (light curves and spectra), and all-sky maps. Illustrative examples of these are provided.