The lack of confirmation for the existence of supersymmetric particles and Weakly Interacting Massive Particles (WIMPs) appeals to extension of the field of studies of the physical nature of dark matter, involving non-supersymmetric and non-WIMP solutions. We briefly discuss some examples of such candidates in their relationship with extension of particle symmetry and pattern of symmetry breaking. We specify in the example of axion-like particles nontrivial features of cosmological reflection of the structure and pattern of Peccei-Quinn-like symmetry breaking. The puzzles of direct and indiect dark matter searches can find solution in the approach of composite dark matter. The advantages and open problems of this approach are specified. We note that detailed analysis of cosmological consequences of any extension of particle model that provides candidates for dark matter inevitably leads to nonstandard features in the corresponding cosmological scenario. It makes possible to use methods of cosmoparticle physics to study physical nature of the dark matter in the combination of its physical, astrophysical and cosmological signatures.
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