  Tremendous ongoing theory efforts are dedicated to developing new methods for
QCD calculations. Qualitative rather than incremental advances are needed to
fully exploit data still to be collected at the LHC. The maximally
supersymmetric Yang-Mills theory (${\mathcal N}=4$ sYM) shares with QCD the
gluon sector, which contains the most complicated Feynman graphs, but at the
same time has many special properties, and is believed to be solvable exactly.
It is natural to ask what we can learn from advances in ${\mathcal N}=4$ sYM
for addressing difficult problems in QCD. With this in mind, we review here
several remarkable developments and highlights of recent results in ${\mathcal
N}=4$ 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, we argue that
there is a host of further fascinating ideas waiting to be explored.
