This work is devoted to the study of three different models of dark matter, which include non-standard interactions with ordinary matter or with itself, and their effects in cosmology and astrophysics. Multi-component dark matter models include more than one sort of new fundamental particles and their interactions and typically have richer phenomenology. The introduction of such additional candidates allows to explain various observational discrepancies in astrophysics or cosmology and/or offers new possibilities to probe the nature of dark matter. The excess of high-energy cosmic positrons above the astrophysical prediction can have a dark matter origin. We show that, however, dark matter explanations of this phenomenon, which assume a conventional density distribution in the Galaxy, are in serious tension with the observations of the isotropic gamma-ray background. As a possible alternative explanation, we investigate the model of self-interacting dark matter, which forms a disk and alleviates the problem with gamma-ray mea- surements. Some models of strongly interacting massive particles (SIMPs) might produce the signal in the DAMA detector and explain the contradiction with the results of other direct detection experiments. We study the propagation of SIMPs in the ground and demonstrate that such models cannot reproduce the time-dependent features of the observed rate of events. The 3-Higgs-Doublet Model (3HDM) with the CP symmetry of order 4 is the simplest extension of the Higgs sector that allows to stabilize potential dark matter candidates without any additional symmetries. This model contains a new inter- action that converts one dark matter candidate into another and leads to novel phenomenology of asymmetric dark matter. We describe the cosmological ther- mal evolution of dark matter density in this model and discuss the perspectives of indirect detection.