A muon collider would enable the big jump ahead in energy reach that is needed for a fruitful exploration of fundamental interactions. The challenges of producing muon collisions at high luminosity and 10 TeV centre of mass energy are being investigated by the recently-formed International Muon Collider Collaboration. This Review summarises the status and the recent advances on muon colliders design, physics and detector studies. The aim is to provide a global perspective of the field and to outline directions for future work.

Run 2 LHC data show hints of a new resonance in the diphoton distribution at an invariant mass of 750 GeV. We analyse the data in terms of a new boson, extracting information on its properties and exploring theoretical interpretations. Scenarios covered include a narrow resonance and, as preliminary indications suggest, a wider resonance. If the width indications persist, the new particle is likely to belong to a strongly-interacting sector. We also show how compatibility between Run 1 and Run 2 data is improved by postulating the existence of an additional heavy particle, whose decays are possibly related to dark matter.

Neutrino masses can be generated by fermion triplets with TeV-scale mass, that would manifest at LHC as production of two leptons together with two heavy standard model (SM) vectors or Higgs, giving rise to final states such as $2\ell +4j$ (that can violate lepton number and/or lepton flavor) or $\ell +4j+{\mathrm{E̸}}_T$. We devise cuts to suppress the SM backgrounds to these signatures. Furthermore, for most of the mass range suggested by neutrino data, triplet decays are detectably displaced from the production point, allowing to infer the neutrino mass parameters. We compare with LHC signals of type-I and type-II seesaw.

We study the conditions for a new scalar resonance to be observed first in diphotons at the LHC Run-2. We focus on scenarios where the scalar arises either from an internal or spacetime symmetry broken spontaneously, for which the mass is naturally below the cutoff and the low-energy interactions are fixed by the couplings to the broken currents, UV anomalies, and selection rules. We discuss the recent excess in diphoton resonance searches observed by ATLAS and CMS at 750 GeV, and explore its compatibility with other searches at Run-1 and its interpretation as Goldstone bosons in super-symmetry and composite Higgs models. We show that two candidates naturally emerge: a Goldstone boson from an internal symmetry with electromagnetic anomalies, and the scalar partner of the Goldstone of supersymmetry breaking: the sgoldstino. The dilaton from conformal symmetry breaking is instead disfavoured by present data, in its minimal natural realization.