Dark matter is a milestone in the understanding of the Universe and a portal to the discovery of new physics beyond the Standard Model of particles. The direct search for dark matter has become one of the most active fields of experimental physics in the last few decades. Liquid Xenon (LXe) detectors demonstrated the highest sensitivities to the main dark matter candidates (Weakly Interactive Massive Particles, WIMP). The experiments of the XENON project, located in the underground INFN Laboratori Nazionali del Gran Sasso (LNGS) in Italy, are leading the field thanks to the dual-phase LXe time projection chamber (TPC) technology. Since the first prototype XENON10 built in 2005, each detector of the XENON project achieved the highest sensitivity to WIMP dark matter. XENON increased the LXe target mass by nearly a factor 400, up to the 5.9 t of the current XENONnT detector installed at LNGS in 2020. Thanks to an unprecedentedly low background level, XENON1T (predecessor of XENONnT) set the world best limits on WIMP dark matter to date, for an overall boost of more than 3 orders of magnitude to the experimental sensitivity since the XENON project started. In this work, we review the principles of direct dark matter detection with LXe TPCs, the detectors of the XENON project, the challenges posed by background mitigation to ultra-low levels, and the main results achieved by the XENON project in the search for dark matter.