The first part of the talk will introduce the two Effective Field Theories that parameterize small effects of heavy Beyond-Standard-Model physics, namely the Standard Model EFT (SMEFT)  and the Higgs EFT (HEFT). It will discuss how their formalisms compare, as well as their relevance for current new physics searches at the LHC. The second part will present a novel geometric formulation aiming at giving a parameterization of the scalar sector of both EFTs including terms up to arbitrarily high number of derivatives, in a way that is independent of the particular representation chosen for the scalar fields. The formulation employs the mathematical notion of jet bundles and was first presented 2308.00017