Direct searches for dark matter lead to serious problems for simple models with stable neutral Weakly Interacting Massive Particles (WIMPs) as candidates for dark matter. A possibility is discussed that new stable quarks and charged leptons exist and are hidden from detection, being bound in neutral dark atoms of composite dark matter. Stable -2 charged particles O^-- are bound with primordial helium in O-helium (OHe) atoms, being specific nuclear interacting form of composite Warmer than Cold dark matter. Slowed down in the terrestrial matter, OHe is elusive for direct methods of underground dark matter detection based on the search for effects of nuclear recoil in WIMP-nucleus collisions. The positive results of DAMA experiments can be explained as annual modulation of radiative capture of O-helium by nuclei. In the framework of this approach, test of DAMA results in detectors with other chemical content becomes a nontrivial task, while the experimental search of stable charged particles at LHC or in cosmic rays acquires a meaning of direct test for composite dark matter scenario.