The Higgs boson provides a unique tool to study the physics of the Standard Model, as well as to search for manifestations of physics beyond it. The first part of this thesis is devoted to precision in the Higgs sector, where we perform a phenomenological analysis of ZZ production at the Large Hadron Collider (LHC) using the recently developed MiNNLOPS method for the matching of fixed-order QCD calculations to a parton shower. In the second part of this thesis, we explore a number of phenomenological directions to search for physics beyond the Standard Model. Within the context of the Standard Model effective field theory, we probe possible modifications of the trilinear Higgs coupling via off-shell Higgs production at the LHC, and use of a variety of non-Higgs observables to constrain dipole-type operators that modify the bottom-Higgs and charm-Higgs dynamics. Finally, we perform a sensitivity study for the high-luminosity upgrade of the LHC and other future collider options in constraining Higgs-portal interactions.