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It probes structure formation over more than 10 billion years primarily from the combination of weak gravitational lensing and galaxy clustering data. The breadth of <jats:italic>Euclid<\/jats:italic>\u2019s data will also foster a wide variety of scientific analyses. The Flagship simulation was developed to provide a realistic approximation to the galaxies that will be observed by <jats:italic>Euclid<\/jats:italic> and used in its scientific exploitation. We ran a state-of-the-art <jats:italic>N<\/jats:italic>-body simulation with four trillion particles, producing a lightcone on the fly. From the dark matter particles, we produced a catalogue of 16 billion haloes in one octant of the sky in the lightcone up to redshift <jats:italic>z<\/jats:italic> = 3. We then populated these haloes with mock galaxies using a halo occupation distribution and abundance-matching approach, calibrating the free parameters of the galaxy mock against observed correlations and other basic galaxy properties. Modelled galaxy properties include luminosity and flux in several bands, redshifts, positions and velocities, spectral energy distributions, shapes and sizes, stellar masses, star formation rates, metallicities, emission line fluxes, and lensing properties. We selected a final sample of 3.4 billion galaxies with a magnitude cut of <jats:italic>H<\/jats:italic><jats:sub>E<\/jats:sub> &lt; 26, where we are complete. We have performed a comprehensive set of validation tests to check the similarity to observational data and theoretical models. In particular, our catalogue is able to closely reproduce the main characteristics of the weak lensing and galaxy clustering samples to be used in the mission main cosmological analysis. 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