Proton-proton collisions at the LHC produce an immense flux of high-energy (TeV-scale) neutrinos along the beam (forward) direction. The breakthrough observation of LHC neutrinos by FASER and SND@LHC in March 2023 and the first measurement of neutrino interactions at the TeV scale by FASERnu herald a new era in particle physics. In this talk I present the unprecedented reach of current (FASER, SND@LHC) and future (FASER2, AdvSND, FLArE) forward LHC neutrino experiments for studies of SM and BSM physics, focusing on their impact for QCD and neutrino physics. I demonstrate that these experiments effectively extend the CERN infrastructure with an Neutrino-Ion Collider, enabling novel opportunities to pin down hadronic structure, reduce theory errors in Higgs cross-sections, enhance the direct search program of the HL-LHC, open a window to the gluon content of the proton in regions outside the coverage of any other experiment, and provide the first laboratory-based validation of cross-sections relevant for high-energy astroparticle experiments.