Context. Gaia has been in operations since 2014, and two full data releases (DR) have been delivered so far: DR1 in 2016 and DR2 in 2018. The third Gaia data release expands from the early data release (EDR3) in 2020, which contained the five-parameter astrometric solution and mean photometry for 1.8 billion sources by providing 34 months of multi-epoch observations that allowed us to systematically probe, characterise, and classify variable celestial phenomena. Aims. We present a summary of the variability processing and analysis of the photometric and spectroscopic time series of 1.8 billion sources carried out for Gaia DR3. Methods. We used statistical and machine learning methods to characterise and classify the variable sources. Training sets were built from a global revision of major published variable star catalogues. For a subset of classes, specific detailed studies were conducted to confirm their class membership and to derive parameters that are adapted to the peculiarity of the considered class. Results. In total, 10.5 million objects are identified as variable in Gaia DR3 and have associated time series in G, G(BP), and G(RP) and, in some cases, radial velocity time series. The DR3 variable sources subdivide into 9.5 million variable stars and 1 million active galactic nuclei or `quasars'. In addition, supervised classification identified 2.5 million galaxies thanks to spurious variability induced by the extent of these objects. The variability analysis output in the DR3 archive amounts to 17 tables, containing a total of 365 parameters. We publish 35 types and subtypes of variable objects. For 11 variable types, additional specific object parameters are published. Here, we provide an overview of the estimated completeness and contamination of most variability classes. Conclusions. Thanks to Gaia, we present the largest whole-sky variability analysis based on coherent photometric, astrometric, and spectroscopic data. Future Gaia data releases will more than double the span of time series and the number of observations, allowing the publication of an even richer catalogue.