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Gaia DR3 contains 1.8 billion sources with G-band photometry, 1.5 billion of which with BP and RP photometry, complemented by positions on the sky, parallax, and proper motion. The median number of field-of-view transits in the three photometric bands is between 40 and 44 measurements per source and covers 34 months of data collection. We pursue a classification of Galactic and extra-galactic objects that are detected as variable by Gaia across the whole sky. Supervised machine learning (eXtreme Gradient Boosting and Random Forest) was employed to generate multi-class, binary, and meta-classifiers that classified variable objects with photometric time series in the G, BP, and RP bands. Classification results comprise 12.4 million sources (selected from a much larger set of potential variable objects) and include about 9 million variable stars classified into 22 variability types in the Milky Way and nearby galaxies such as the Magellanic Clouds and Andromeda, plus thousands of supernova explosions in distant galaxies, 1 million active galactic nuclei, and almost 2.5 million galaxies. The identification of galaxies was made possible by the artificial variability of extended objects as detected by Gaia, so they were published in the galaxy_candidates table of the Gaia DR3 archive, separate from the classifications of genuine variability (in the vari_classifier_result table). The latter contains 24 variability classes or class groups of periodic and non-periodic variables (pulsating, eclipsing, rotating, eruptive, cataclysmic, stochastic, and microlensing), with amplitudes from a few milli-magnitudes to several magnitudes.