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.
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