Observations on all fronts strongly support the view of a universe composed of >96% invisible matter and energy. The invisible matter is non-baryonic, cold and likely in the form of new particles generically referred to as Weakly Interacting Massive Particles (WIMPs), relics from the early universe. One way to detect WIMPs is to measure the nuclear recoils produced in their rare elastic collisions with ordinary matter. The predicted interaction rate ranges from the best sensitivity of existing experiments of ~1 evts/kg/yr to ~1 evts/1000 kg/yr. Efforts are underway worldwide to realize sensitive direct detection experiments, with large target mass and improved background rejection capabilities. In this talk I will review experiments headed in this direction with the use of cryogenic noble liquids, focusing on those experiments which use the common technique of a dual-phase (liquid/gas) time projection chamber to measure simultaneously the ionization and the scintillation signals produced by radiation in a large volume of liquid xenon or liquid argon. These include experiments such as XENON, ZEPLIN, WARP and ArDM.