The LHeC and the FCC-eh offer fascinating, unique possibilities for discovering BSM physics in DIS, both due to their large centre-of-mass energies and high luminosities. In this talk we will review most recent studies as presented in the 2020 LHeC White Paper. We will show the prospects for observing extensions of the Higgs sectors both with charged and neutral scalars, anomalous Higgs couplings and exotic decays. Then we will discuss searches for $R$-parity conserving and violating supersymmetry with both prompt and long-lived particles, and of feeble interacting particles like sterile neutrinos, fermion triplets, dark photons and axion-like particles. Finally we will address anomalous couplings and searches for heavy resonances like leptoquarks and vector-like quarks, excited fermions and colour-octet leptons.