  We compute the two-loop Quantum Chromodynamics (QCD) corrections to all
partonic channels relevant for the production of an electroweak boson
$V=Z,W^\pm,\gamma^*$ and a jet at hadron colliders. We consider the decay of a
vector boson $V$ to three partons $ V \to q\bar{q}g$, $ V \to ggg$ with a
vector and axial vector coupling in both channels, including singlet and
non-singlet contributions. For the quark channel, we use a recent tensor
decomposition and extend the calculation to $\mathcal{O}(\epsilon^2)$. For the
gluonic channel, we define a new tensor decomposition which allows us to
compute the vector and the axial vector amplitudes at once and to perform the
computation of the amplitudes to $\mathcal{O}(\epsilon^2)$. We provide finite
remainders of the helicity amplitudes analytically continued to all relevant
scattering regions $q\bar{q} \to V g$, $q g \to V q$ and $gg \to V g$. The
axial vector contribution to the gluon-induced channel completes the set of
two-loop amplitudes for this process, while the extension to
$\mathcal{O}(\epsilon^2)$ represents the first step in the calculation of
next-to-next-to-next-to-leading-order (N$^3$LO) QCD corrections to $Z$+jet
production at hadron colliders.
