We present EMERGE, an Empirical ModEl for the foRmation of GalaxiEs, describing the evolution of individual galaxies in large volumes from $zsim10$ to the present day. We assign a star formation rate to each dark matter halo based on its growth rate, which specifies how much baryonic material becomes available, and the instantaneous baryon conversion efficiency, which determines how efficiently this material is converted to stars, thereby capturing the baryonic physics. Satellites are quenched following the delayed-then-rapid model, and they are tidally disrupted once their subhalo has lost a significant fraction of its mass. The model is constrained with observed data extending out to high redshift. The empirical relations are very flexible, and the model complexity is increased only if required by the data, assessed by several model selection statistics. We find that for the same final halo mass galaxies can have very different star formation histories. Nevertheless, the average star formation and accretion rates are in good agreement with models following an abundance matching strategy. Galaxies that are quenched at $z=0$ typically have a higher peak star formation rate compared to their star-forming counterparts. The accretion of stars can dominate the total mass of massive galaxies, but is insignificant for low-mass systems, independent of star-formation activity. EMERGE predicts stellar-to-halo mass ratios for individual galaxies and introduces scatter self-consistently. We find that at fixed halo mass, passive galaxies have a higher stellar mass on average. The intra-cluster-mass in massive haloes can be up to 8 times larger than the mass of the central galaxy. Clustering for star-forming and quenched galaxies is in good agreement with observational constraints, indicating a realistic assignment of galaxies to haloes.
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