A new era in particle physics is being spurred on by new data from the Large Hadron Collider. Non-vanishing neutrino masses represent firm observational evidence of new physics beyond the Standard Model. An extension of the latter, based on a SU(3)_C × SU(2)_L × U(1)_Y × U(1)_B-L symmetry, incorporating an established Baryon minus Lepton number invariance, is proposed as a viable and testable solution to the neutrino mass problem. We argue that LHC data will probe all the new content of this model: heavy neutrinos, an extra gauge boson emerging from spontaneous breaking of the additional gauge group at the TeV scale, onset by a new heavier Higgs boson, also visible at the CERN proton-proton collider. An even more exciting version of this model is the one exploiting Supersymmetry: firstly, it incurporates all its well-known benefits; secondly, it alleviates the flaws of its more minimal realisations. Finally, this model provides a credible cold Dark Matter candidate, the lightest sneutrino, detectable in both underground and collider experiments.