  We study the phenomenological consequences of several $CP$-violating
structures that could arise in the Standard Model effective field theory
framework. Focusing on operators involving electroweak gauge and/or Higgs
bosons, we derive constraints originating from Run I LHC data. We then study
the capabilities of the present and future LHC runs at higher energies to
further probe associated CP-violating phenomena and we demonstrate how
differential information can play a key role. We consider both traditional
four-lepton probes of CP-violation in the Higgs sector and novel new physics
handles based on varied angular and non-angular observables.
