  Measurements of production cross sections of WZ and same-sign WW boson pairs
in association with two jets in proton-proton collisions at $\sqrt{s} =$ 13 TeV
at the LHC are reported. The data sample corresponds to an integrated
luminosity of 137 fb$^{-1}$, collected with the CMS detector during 2016-2018.
The measurements are performed in the leptonic decay modes W$^\pm$Z $\to$
$\ell^\pm\nu\ell'^\pm\ell'^\mp$ and WW $\to$ $\ell^\pm\nu\ell'^\pm\nu$, where
$\ell, \ell' = $ e, $\mu$. Differential fiducial cross sections as functions of
the invariant masses of the jet and charged lepton pairs, as well as of the
leading-lepton transverse momentum, are measured for WW production and are
consistent with the standard model predictions. The dependence of differential
cross sections on the invariant mass of the jet pair is also measured for WZ
production. An observation of electroweak production of WZ boson pairs is
reported with an observed (expected) significance of 6.8 (5.3) standard
deviations. Constraints are obtained on the structure of quartic vector boson
interactions in the framework of effective field theory.
