With the discovery of the Higgs at the LHC, experiments have finally addressed all aspects of the Standard Model (SM). At this stage, it is important to understand which windows for beyond the SM (BSM) physics are still open, and which are instead tightly closed. We address this question by parametrizing BSM effects with dimension-six operators and performing a global fit to the SM. We separate operators into different groups constrained at different levels, and provide independent bounds on their Wilson coefficients taking into account only the relevant experiments. Our analysis allows to assert in a model-independent way where BSM effects can appear in Higgs physics. In particular, we show that deviations from the SM in the differential distributions of h -> V (f) over bar f are related to other observables, such as triple gauge-boson couplings, and are then already constrained by present data. On the contrary, BR(h -> Z gamma) can still hide large deviations from the SM.