With the discovery of the Higgs boson the LHC has confirmed an incredible prediction of the Standard Model precision program and opened new horizons towards the even more ambitious goal of constraining new physics through precision Higgs-boson physics. With this respect, the capabilities of the LHC as a precision physics machine should not be underestimated. The progress of experimental analyses has been matched by an unprecedented theoretical effort to describe both signals (Higgs, new physics) and backgrounds (SM). In most cases the measurement of Higgs production and properties is not limited these days by theoretical systematic, but cases still exist where this is the case. In these cases, further effort to reach a more satisfactory theoretical accuracy will have to be matched by a dedicated program of experimental measurements. In this talk I will review the interplay between theory and experiments in defining a Higgs precision-physics program, and I will discuss how, when combined with global electroweak precision fits, this can be used to constrain extensions of the Standard Model.