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One of the main goals of the LHC is the search for New Physics. So far, the only new particle that has been discovered at the LHC is the Higgs boson. There is no direct sign of any particle beyond the Standard Model (SM) so far. In view of this situation we have to ask ourselves how we can exploit the Higgs sector to search indirectly for New Physics. I will discuss the role that the effective field theory (EFT) approach plays in this context. EFT is a possibility to smoothly depart from the SM and parametrize our ignorance about NP by assuming that NP appears at some high energy scale. I will focus on the top-Higgs interface and Higgs pair production and discuss the role of global fits. Since EFT assumes NP to be heavy, the combined analysis within EFT and specific UV-complete models is important in order not to miss any sign of NP. I will hence also present some specific examples of Higgs pair production in UV-complete models.