  Electron-proton ($e^-p$) colliders are an ideal laboratory to study common
features of electron and quarks with production via electroweak bosons,
leptoquarks, multi-jet final states and very forward physics, due to their
impressive pseudorapidity coverage.
  In addition to these physics cases, there exist a broad Beyond the Standard
Model (BSM) program aimed at exploring the capabilities of the LHeC [1] and
FCC-he [2] for several New Physics scenarios. Although their centre-of-mass
energy is down with respect to a $pp$ collider by a factor of $\sqrt{E_p/E_e}
\sim 10~(30)$ for the LHeC (FCC-he), they can be an invaluable tool to
characterize BSM physics hints at $ee$ and $pp$ machines.
  The aim of this talk is to provide, on behalf of the BSM $e^-p$ Working
Group, an overview of the aforementioned BSM program, by briefly summarizing
the existing studies and reporting on the most recent progress. We expect that
the ample scope in terms of NP models to be tested would enhance the synergies
between the BSM and $e^-p$ communities
