  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.
