We give a brief review of beyond-the-Standard Model (BSM) extensions of the Standard Model (SM) Higgs sector. Going from very simple to more complicated models, our survey includes models with additional scalar singlet fields, the Two Higgs Doublet Model (2HDM) and the Higgs sector of the Minimal Supersymmetric Standard Model (MSSM). We discuss the impact of current experimental results from LHC Higgs searches and measurements and the prospective reach of the LHC in the high-luminosity (HL) phase. We furthermore highlight possible new collider signatures within these models that have not been experimentally probed to date.