  We present a parton-level study of electro-weak production of vector-boson
pairs at the Large Hadron Collider, establishing the sensitivity to a set of
dimension-six operators in the Standard Model Effective Field Theory (SMEFT).
Different final states are statistically combined, and we discuss how the
orthogonality and interdependence of different analyses must be considered to
obtain the most stringent constraints. The main novelties of our study are the
inclusion of SMEFT effects in non-resonant diagrams and in irreducible QCD
backgrounds, and an exhaustive template analysis of optimal observables for
each operator and process considered. We also assess for the first time the
sensitivity of vector-boson-scattering searches in semileptonic final states.
