  Measurements of integrated and differential cross-sections for electroweak
$W^{\pm}Z$ production in association with two jets ($W^{\pm}Zjj$) in
proton-proton collisions are presented. The data collected by the ATLAS
detector at the Large Hadron Collider from $2015$ to $2018$ at a centre-of-mass
energy of $\sqrt{s} = 13$ TeV are used, corresponding to an integrated
luminosity of $140$ fb$^{-1}$. The $W^{\pm}Zjj$ candidate events are
reconstructed using leptonic decay modes of the gauge bosons. Events containing
three identified leptons, either electrons or muons, and two jets are selected.
Processes involving pure electroweak $W^{\pm}Zjj$ production at Born level are
separated from $W^{\pm}Zjj$ production involving a strong coupling. The
measured integrated fiducial cross-section of electroweak $W^{\pm}Zjj$
production per lepton flavour is $\sigma_{WZjj\mathrm{-EW} \rightarrow \ell^{'}
\nu \ell \ell jj} = 0.368 \; \pm 0.037 \,(\mathrm{stat.}) \; \pm 0.059
\,(\mathrm{syst.}) \; \pm 0.003 \,(\mathrm{lumi.}) \; \mathrm{fb}$. Respective
cross-sections of electroweak and strong $W^{\pm}Zjj$ production are measured
separately for events with exactly two jets or with more than two jets, and in
three bins of the invariant mass of the two jets. The inclusive $W^{\pm}Zjj$
production cross-section, without separating electroweak and strong production,
is also measured to be $\sigma_{WZjj \rightarrow \ell^{'} \nu \ell \ell jj} =
1.462 \; \pm 0.063 \,(\mathrm{stat.}) \; \pm 0.118 \,(\mathrm{syst.}) \; \pm
0.012 \,(\mathrm{lumi.}) \; \mathrm{fb}$, per lepton flavour. The inclusive
$W^{\pm}Zjj$ production cross-section is measured differentially for several
kinematic observables. Finally, the measurements are used to constrain
anomalous quartic gauge couplings by extracting $95$ % confidence level
intervals on dimension-$8$ operators.
