Context. The third data release by the Gaia mission of the European Space Agency (DR3) is the first release to provide the community with a large sample of observations for more than 150 thousand Solar System objects, including asteroids and natural planetary satellites. The release contains astrometry (over 23 million epochs) and photometry, along with average reflectance spectra of 60518 asteroids and osculating elements. Aims. We present an overview of the procedures that have been implemented over several years of development and tests to process Solar System data at the level of accuracy that Gaia can reach. We illustrate the data properties and potential with some practical examples. Methods. In order to allow the users of Gaia DR3 to best exploit the data, we explain the assumptions and approaches followed in the implementation of the data processing pipeline for Solar System processing, and their effects in terms of data filtering, optimisation, and performances. We then test the data quality by analysing post-fit residuals to adjusted orbits, the capacity of detecting subtle dynamical effects (wobbling due to satellites or shape and Yarkovsky acceleration), and to reproduce known properties of asteroid photometry (phase curves and rotational light curves). Results. The DR3 astrometric accuracy is a clear improvement over the data published in DR2, which concerned a very limited sample of asteroids. The performance of the data reduction is met, and is illustrated by the capacity of detecting milliarcsecond-level wobbling of the asteroid photocentre that is due to satellite or shape effects and contributes to Yarkovsky effect measurements. Conclusions. The third data release can in terms of data completeness and accuracy be considered the first full-scale realisation of the Solar System survey by Gaia.