  SHiP is a newly proposed fixed-target experiment at the CERN SPS with the aim
of searching for hidden particles that interact very weakly with SM particles.
The work presented in this document investigates SHiP's physics reach in the
parameter space of the Neutrino Minimal Standard Model ($\nu$MSM), which is a
theory that could solve most problems left open by the Standard Model with
sterile neutrinos. A model introducing an extra $U(1)$ symmetry in the hidden
sector, providing a natural candidate for dark matter, is also explored. This
work shows that the SHiP experiment can improve by several orders of magnitude
the sensitivity to Heavy Neutral Leptons below 2 GeV, scanning a large part of
the parameter space below the $B$ meson mass. The remainder of the $\nu$MSM
parameter space, dominated by right-handed neutrinos with masses above 2 GeV,
can be explored at a future $e^+e^-$ collider. Similarly, SHiP can greatly
improve present constraints on $U(1)$ dark photons.
