<jats:p> The existence of cosmological dark matter is in the bedrock of the modern cosmology. The dark matter is assumed to be nonbaryonic and consists of new stable particles. Weakly Interacting Massive Particle (WIMP) miracle appeals to search for neutral stable weakly interacting particles in underground experiments by their nuclear recoil and at colliders by missing energy and momentum, which they carry out. However, the lack of WIMP effects in their direct underground searches and at colliders can appeal to other forms of dark matter candidates. These candidates may be weakly interacting slim particles, superweakly interacting particles, or composite dark matter, in which new particles are bound. Their existence should lead to cosmological effects that can find probes in the astrophysical data. However, if composite dark matter contains stable electrically charged leptons and quarks bound by ordinary Coulomb interaction in elusive dark atoms, these charged constituents of dark atoms can be the subject of direct experimental test at the colliders. The models, predicting stable particles with charge [Formula: see text] without stable particles with charges [Formula: see text] and [Formula: see text] can avoid severe constraints on anomalous isotopes of light elements and provide solution for the puzzles of dark matter searches. In such models, the excessive [Formula: see text] charged particles are bound with primordial helium in O-helium atoms, maintaining specific nuclear-interacting form of the dark matter. The successful development of composite dark matter scenarios appeals for experimental search for doubly charged constituents of dark atoms, making experimental search for exotic stable double charged particles experimentum crucis for dark atoms of composite dark matter. </jats:p>