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The third Gaia Data Release, covering 34 months of data, includes the second Gaia catalogue of long-period variables (LPVs), with G variability amplitudes larger than 0.1 mag (5-95% quantile range). The paper describes the production and content of this catalogue, and the methods used to compute the published variability parameters and identify C-star candidates. We applied various filtering criteria to minimise contamination by other kinds of variables. The variability parameters, period and amplitude, were derived from model fits to the G-band light curves, wherever possible. C stars were identified using their molecular signature in the low-resolution RP spectra. The catalogue contains 1 720 558 LPV candidates, including 392 240 stars with published periods (ranging from 35 to1000 days) and 546 468 stars classified as C-stars candidates. Comparison with literature data (OGLE and ASAS-SN) leads to an estimated 80% of completeness. The recovery rate is about 90% for the most regular stars (typically Miras) and 60% for semi-regular and irregular ones. At the same time, the number of known LPVs is increased by a large factor with respect to the literature data, especially in crowded regions, and the contamination is estimated to be below two percents. Our C-star classification, based on solid theoretical arguments, is consistent with spectroscopically identified C stars in the literature. Caution must however be taken if the S/N ratio is small, in crowded regions or if the source is reddened by some kind of extinction. The quality and potential of the catalogue are illustrated by presenting and discussing LPVs in the solar neighbourhood, in globular clusters and in galaxies of the Local Group. This is the largest all-sky catalogue of LPVs to date with a photometric depth down to G=20 mag, providing a unique data set for research on late stages of stellar evolution.