The puzzling Higgs: physics beyond the Standard Model after the first LHC phase

The first three-year run of the LHC with up to 8 TeV center-of-mass energy concluded in 2012. The greatest achievement was the discovery of the Higgs boson which, so far, is perfectly compatible with the properties predicted by the Standard Model. Although strong theoretical arguments imply that the Higgs boson should not exist in isolation, no sign of physics beyond the Standard Model has been discovered. This talk will review the theoretical motivation for and the status of two of the most well-motivated models for a "natural" Higgs boson, supersymmetry and Higgs compositeness, and their prospects for the upcoming 13 TeV LHC run.