A little over a decade after the discovery of the Higgs boson at the Large Hadron Collider, a microscopic understanding of the nature of the electroweak scale remains elusive. This is particularly perplexing given that the Standard Model of Particle Physics, successful as it is, cannot be the ultimate description of nature. Perhaps, then, the challenge is not only to gather more data but to reconsider how we interpret it. In this talk, I will discuss ways to move, with theoretical precision, away from constraining phenomenological patterns of the electroweak scale as predicted by the SM or even its effective generalisation. I show how such an approach, inspired by geometry and effective field theory, can inform the search for physics beyond the Standard Model via the Higgs boson’s interactions. These chart possibilities for beyond-the-SM particle physics at the TeV scale, with direct relevance for the present and future (collider) physics programme. Such approaches are also more broadly relevant in other phenomenological arenas, offering new directions for exploring the potential connections among the electroweak scale, theories of dark matter, phase transitions in the early Universe, and Inflation.