In this talk, I will review the possibility that a Dark Matter candidate
lives on a 4-dimensional plane (a "brane") located in a 5-dimensional
space-time, with an extra spatial dimension compactified either in
Randall-Sundrum or in Clockwork/Linear Dilaton geometries. The DM
particle only interacts with SM particles through gravity, whose
couplings are enhanced with respect to the standard 4D-case due to the
extra-dimensional framework. In order to achieve the observed DM relic
abundance $\Omega_{\rm DM}$, we considered both the Freeze-Out and the
Freeze-In mechanisms. We have worked out the regions in the
corresponding parameter space for which $\Omega_{\rm DM}$ may be
achieved, once bounds from
the LHC and Direct and Indirect Dark Matter searches are taken into account.
I will also discuss briefly why an extra-dimensional scenario may be
appealing and give a short outlook of future developments.