In recent years the use of effective field theory techniques has come to the fore when studying the LHC data set. This approach constrains the possible effects of physics beyond the Standard Model, by constraining small modifications of Standard Model interactions in a theory known as the Standard Model Effective Field Theory. Although highly practical and agnostic, the presence of a Higgs field that takes on a background expectation value leads to some unique challenges in formulating this effective field theory and interfacing the SMEFT with the data. We will review some of the successes of this approach and some of the problems that have appeared. A growing understanding of the physics of the SMEFT as emerging from the geometry of Higgs field space offers great potential to tackle some of the remaining challenges. We will also introduce and discuss this approach.