Higgs Supersymmetry and Dark Matter After Run I of the LHC Introduction At Least a Higgs Boson Interpreting LHC Searches for New Physics Conclusions. . This work was nominated as an outstanding PhD thesis by the LPSC Université Grenoble Alpes France. The LHC Run 1 was a milestone in particle physics leading to the discovery of the Higgs boson the last missing piece of the so called Standard Model (SM) and to important constraints on new physics which challenge popular theories like weak scale supersymmetry. This thesis provides a detailed account of the legacy of the LHC Run 1 ≤¥regarding these aspects. First the SM and the need for its extension are presented in a concise yet revealing way. Subsequently the impact of the LHC Higgs results on scenarios of new physics is assessed in detail including a careful discussion of the relevant uncertainties. Two approaches are considered: generic modifications of the Higgs couplings possibly arising from extended Higgs sectors or higher dimensional operators; and tests of specific new physics models. Lastly the implications of the null results of the searches for new physics are discussed with a particular focus on supersymmetric dark matter candidates. Here as well two approaches are presented: the simplified models approach and recasting by event simulation. This thesis stands out for its educational approach its clear language and the depth of the physics discussion. The methods and tools presented offer readers essential practical tools for future research. .