  We report on the first fully differential calculation for $W^\pm Z$
production in hadron collisions up to next-to-next-to-leading order (NNLO) in
QCD perturbation theory. Leptonic decays of the $W$ and $Z$ bosons are
consistently taken into account, i.e. we include all resonant and non-resonant
diagrams that contribute to the process $pp\to \ell^{'\pm} \nu_{\ell^{'}}
\ell^+\ell^-+X$ both in the same-flavour ($\ell'=\ell$) and the
different-flavour ($\ell'\neq \ell$) channel. Fiducial cross sections and
distributions are presented in the presence of standard selection cuts applied
in the experimental $W^\pm Z$ analyses by ATLAS and CMS at centre-of-mass
energies of 8 and 13\,TeV. As previously shown for the inclusive cross section,
NNLO corrections increase the NLO result by about $10\%$, thereby leading to an
improved agreement with experimental data. The importance of NNLO accurate
predictions is also shown in the case of new-physics scenarios, where,
especially in high-$p_T$ categories, their impact can reach ${\cal O}(20\%)$.
The availability of differential NNLO predictions will play a crucial role in
the rich physics programme that is based on precision studies of $W^\pm Z$
signatures at the LHC.
