We review scenarios in which the particles that account for the Dark Matter (DM) in the Universe interact only through their couplings with the Higgs sector of the theory, the so-called Higgs-portal models. In a first step, we use a general and model-independent approach in which the DM particles are singlets with spin $0,frac12$ or $1$, and assume a minimal Higgs sector with the presence of only the Standard Model (SM) Higgs particle observed at the LHC. In a second step, we discuss non-minimal scenarios in which the spin-$frac12$ DM particle is accompanied by additional lepton partners and consider several possibilities like sequential, singlet-doublet and vector-like leptons. In a third step, we examine the case in which it is the Higgs sector of the theory which is enlarged either by a singlet scalar or pseudoscalar field, an additional two Higgs doublet field or by both; in this case, the matter content is also extended in several ways. Finally, we investigate the case of supersymmetric extensions of the SM with neutralino DM, focusing on the possibility that the latter couples mainly to the neutral Higgs particles of the model which then serve as the main portals for DM phenomenology. In all these scenarios, we summarize and update the present constraints and future prospects from the collider physics perspective, namely from the determination of the SM Higgs properties at the LHC and the search for its invisible decays into DM, and the search for heavier Higgs bosons and the DM companion particles at high-energy colliders. We then compare these results with the constraints and prospects obtained from the cosmological relic abundance as well as from direct and indirect DM searches in astroparticle physics experiments. The complementarity of collider and astroparticle DM searches is investigated in all the considered models.
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