We review the present status of the Standard Model calculation of the anomalous magnetic moment of the muon. This is performed in a perturbative expansion in the fine-structure constant α and is broken down into pure QED, electroweak, and hadronic contributions. The pure QED contribution is by far the largest and has been evaluated up to and including O(α5) with negligible numerical uncertainty. The electroweak contribution is suppressed by (mμ∕MW)2 and only shows up at the level of the seventh significant digit. It has been evaluated up to two loops and is known to better than one percent. Hadronic contributions are the most difficult to calculate and are responsible for almost all of the theoretical uncertainty. The leading hadronic contribution appears at O(α2) and is due to hadronic vacuum polarization, whereas at O(α3) the hadronic light-by-light scattering contribution appears. Given the low characteristic scale of this observable, these contributions have to be calculated with nonperturbative methods, in particular, dispersion relations and the lattice approach to QCD. The largest part of this review is dedicated to a detailed account of recent efforts to improve the calculation of these two contributions with either a data-driven, dispersive approach, or a first-principle, lattice-QCD approach. The final result reads aμSM=116591810(43)×10−11 and is smaller than the Brookhaven measurement by 3.7σ. The experimental uncertainty will soon be reduced by up to a factor four by the new experiment currently running at Fermilab, and also by the future J-PARC experiment. This and the prospects to further reduce the theoretical uncertainty in the near future – which are also discussed here – make this quantity one of the most promising places to look for evidence of new physics.

Motivated by the success of the flavour physics programme carried out over the last decade at the Large Hadron Collider (LHC), we characterize in detail the physics potential of its High-Luminosity and High-Energy upgrades in this domain of physics. We document the extraordinary breadth of the HL/HE-LHC programme enabled by a putative Upgrade II of the dedicated flavour physics experiment LHCb and the evolution of the established flavour physics role of the ATLAS and CMS general purpose experiments. We connect the dedicated flavour physics programme to studies of the top quark, Higgs boson, and direct high-$p_T$ searches for new particles and force carriers. We discuss the complementarity of their discovery potential for physics beyond the Standard Model, affirming the necessity to fully exploit the LHC's flavour physics potential throughout its upgrade eras.