  The investigation of weak bosons $V$ ($V=\mathrm{W}^{\pm}$, $\mathrm{Z}$)
produced with or without associated hard QCD jets will be of great
phenomenological interest at the LHC. Owing to the large cross sections and the
clean decay signatures of the vector bosons, weak-boson production can be used
to monitor and calibrate the luminosity of the collider, to constrain the PDFs,
or to calibrate the detector. Moreover, the $Z$+jet(s) final state constitutes
an important background to a large variety of signatures of physics beyond the
Standard Model. To match the excellent experimental accuracy that is expected
at the LHC, we have worked out a theoretical next-to-leading-order analysis of
$V$+jet production at hadron colliders. The focus of this talk will be on new
results on the full electroweak corrections to $Z(\to l^-l^+)$+jet production
at the LHC. All off-shell effects are included in our approach, and the finite
lifetime of the $Z$ boson is consistently accounted for using the complex-mass
scheme. In the following, we briefly introduce the calculation and discuss
selected phenomenological implications of our results.
