Higgs discovery: the beginning or the end of natural EWSB?

We use global fits to analyze the most recent Higgs data from ATLAS, CMS and Tevatron and compare the Standard Model (SM) prediction with natural extensions of the SM. In particular we study wide classes of composite Higgs models based on different coset structures (leading at low energy to different Higgs sectors including extra singlets and Higgs doublets) and different coupling structures of the elementary fermions to the strong sector. We point out in what situations the composite models could improve (or worsen) the fit to the data and compare with similar trends in the MSSM.