Over the last decades, the evidence for existence of a significant component of matter in the Universe, not visible by conventional telescopes, i. e. Dark Matter (DM) has become indisputable. Present measurements indicate that only a few percent of the total matter content in the Universe is made of known particles. The nature of DM is completely unknown, though many theoretically motivated candidates have been proposed, the most popular being a new type of elementary particle created in the early Universe, dubbed Weakly Interacting Massive particle (WIMP). Since 2015, my group has beenparticipating, holding key responsibilities, in the world-wide most sensitive dark matter detection experiment, the xenon based time projection chamber, XENON1T. The idea is to detect the rare scatters of WIMPs on xenon atoms in a deep-underground detector. The collaboration is currently in the process of constructing its successor, XENONnT, which will have excellent discovery potential for dark matter conforming to the WIMP paradigm. The main purpose of this proposal is to allowed us to be in the forefront of the possible discovery of WIMPs. In particular, we envisage to be responsible for the final statistical analysis of the data leading to either a discovery claim or a model exclusion, and exploit the capabilities of the XENONnT TPC to look for alternative dark matter models, that enjoy increasing popularity recently.