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.

We present an update of the Universal FeynRules Output model format, commonly known as the UFO format, that is used by several automated matrix-element generators and high-energy physics software. We detail different features that have been proposed as extensions of the initial format during the last ten years, and collect them in the current second version of the model format that we coin the Universal Feynman Output format. Following the initial philosophy of the UFO, they consist of flexible and modular additions to address particle decays, custom propagators, form factors, the renormalisation group running of parameters and masses, and higher-order quantum corrections.

The mechanism of electroweak symmetry breaking in Little Higgs Models is analyzed in an effective field theory approach. This enables us to identify observable effects irrespective of the specific structure and content of the heavy degrees of freedom. We parameterize these effects in a common operator basis and present the complete set of anomalous contributions to gauge-boson, Higgs, and fermion couplings. If the hypercharge assignments of the model retain their standard form, electroweak precision data are affected only via the S and T parameters and by contact interactions. As a proof of principle, we apply this formalism to the minimal model and consider the current constraints on the parameter space. Finally, we show how the interplay of measurements at LHC and a Linear Collider could reveal the structure of these models.

Weak vector-boson $W,Z$ scattering at high energy probes the Higgs sector and is most sensitive to any new physics associated with electroweak symmetry breaking. We show that in the presence of the 125 GeV Higgs boson, a conventional effective-theory analysis fails for this class of processes. We propose to extrapolate the effective-theory ansatz by an extension of the parameter-free $K$-matrix unitarization prescription, which we denote as direct $T$-matrix unitarization. We generalize this prescription to arbitrary nonperturbative models and describe the implementation as an asymptotically consistent reference model matched to the low-energy effective theory. We present exemplary numerical results for full six-fermion processes at the LHC.

Extensions of the Standard Model of elementary particle physics to noncommutative geometries have been proposed as a low-energy limit of string models. Independent of this motivation, one may consider such a model as an effective field theory with higher-dimensional operators containing an antisymmetric rank-two background field. We study the signals of such a Noncommutative Standard Model (NCSM) and analyze the discovery potential of a future photon collider, considering angular distributions in fermion pair production.