The Higgs boson is an elementary scalar particle, confirmed in 2012 at the LHC, which completes the Standard Model (SM) of particle physics. The ATLAS and CMS collaborations are currently stress testing the Higgs sector of the SM by precisely measuring the properties of this boson. With the data collected at the LHC to date, the evidence for an energy gap between the SM and new physics beyond the SM has grown strong. However, new physics can still be probed even when new particles are too heavy to be directly produced by collisions, as these particles may modify the interactions of the already-known SM particles. Therefore instead of looking for new particles, physicists can look for new types of interactions not present in the SM. The theoretical framework known as Effective Field Theory (EFT) may be used to compute the potential impact of undiscovered heavy particles on the various SM interactions. As such precision measurements of Higgs boson processes offer an important indirect probe of heavy new physics that is complementary to direct searches for new particles. In this seminar we discuss a number of Higgs boson measurements and their interpretation in terms of the tensor structure of Higgs boson interactions using the EFT approach. Prospects for the next LHC data-taking periods, Run 3 and High Luminosity (HL) LHC, are also discussed.