The ESA Gaia mission provides a unique time-domain survey for more than one billion sources brighter than G=20.7 mag. Gaia offers the unprecedented opportunity to study variability phenomena in the Universe thanks to multi-epoch G-magnitude photometry in addition to astrometry, blue and red spectro-photometry, and spectroscopy. Within the Gaia Consortium, Coordination Unit 7 has the responsibility to detect variable objects, classify them, derive characteristic parameters for specific variability classes, and provide global descriptions of variable phenomena. We describe the variability processing and analysis that we plan to apply to the successive data releases, and we present its application to the G-band photometry results of the first 14 months of Gaia operations that comprises 28 days of Ecliptic Pole Scanning Law and 13 months of Nominal Scanning Law. Out of the 694 million, all-sky, sources that have calibrated G-band photometry in this first stage of the mission, about 2.3 million sources that have at least 20 observations are located within 38 degrees from the South Ecliptic Pole. We detect about 14% of them as variable candidates, among which the automated classification identified 9347 Cepheid and RR Lyrae candidates. Additional visual inspections and selection criteria led to the publication of 3194 Cepheid and RR Lyrae stars, described in Clementini et al. (2016). Under the restrictive conditions for DR1, the completenesses of Cepheids and RR Lyrae stars are estimated at 67% and 58%, respectively, numbers that will significantly increase with subsequent Gaia data releases. Data processing within the Gaia Consortium is iterative, the quality of the data and the results being improved at each iteration. The results presented in this article show a glimpse of the exceptional harvest that is to be expected from the Gaia mission for variability phenomena. [abridged]
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