DARk matter WImp search with liquid xenoN (DARWIN) is a future experiment for the direct detection of dark matter based on a multi-ton liquid xenon time projection chamber. The main goal of DARWIN is to reach an unprecedented sensitivity to WIMP (Weakly Interacting Massive Particle)-nuclear recoil cross sections for a large WIMP mass range, down to the level where the irreducible background from neutrino interactions with the target is dominant. This ultra-low background experiment will also have competitive sensitivity to test other dark matter hypotheses such as dark photons and superWIMPs and to detect other rare events like solar neutrinos, neutrinoless double-beta decay of $^{136}$Xe, solar axions and neutrinos from galactic supernovae. In this paper we present the design of the DARWIN detector and the estimated WIMP sensitivity based on the expected backgrounds. Other rare event searches are also discussed.