What Can We Learn About QCD and Collider Physics from $\mathcal {N}=4$ Super Yang–Mills?

Tremendous ongoing theory efforts are dedicated to developing new methods for quantum chromodynamics (QCD) calculations. Qualitative rather than incremental advances are needed to fully exploit data that are still to be collected at the LHC. The maximally supersymmetric Yang–Mills theory, 𝒩=4 super Yang–Mills (sYM), shares with QCD the gluon sector, which contains the most complicated Feynman graphs but also has many special properties and is believed to be solvable exactly. It is natural to ask what we can learn from advances in 𝒩=4 sYM for addressing difficult problems in QCD. With this in mind, I review several remarkable developments and highlights of recent results in 𝒩=4 sYM. This includes all-order results for certain scattering amplitudes, novel symmetries, surprising geometrical structures of loop integrands, novel tools for the calculation of Feynman integrals, and bootstrap methods. While several insights and tools have already been carried over to QCD and have contributed to state-of-the-art calculations for LHC physics, I argue that there is a host of further fascinating ideas waiting to be explored. Expected final online publication date for the Annual Review of Nuclear and Particle Science, Volume 71 is September 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.