We present the first combination of NNLO QCD and NLO EW corrections for vector-boson pair production at the LHC. We consider all final states with two, three and four charged leptons, including resonant and non-resonant diagrams, spin correlations and off-shell effects. Detailed predictions are discussed for three representative channels corresponding toW+W−,W±ZandZZproduction. Both QCD and EW corrections are very significant, and the details of their combination can play a crucial role to achieve the level of precision demanded by experimental analyses. In this context we point out nontrivial issues that arise at large transverse momenta, where the EW corrections are strongly enhanced by Sudakov logarithms and the QCD corrections can feature so-called giant K-factors. Our calculations have been carried out in the Matrix+OpenLoops framework and can be extended to the production of an arbitrary colour singlet in hadronic collisions, provided that the required two-loop QCD amplitudes are available. Combined NNLO QCDand NLO EW predictions for the full set of massive diboson processes will be made publicly available in the next release of Matrix and will be instrumental in advancing precision diboson studies and new-physics searches at the LHC and future hadron colliders.