  The LHC experiments will achieve percent level precision measurements of
processes key to some of the most pressing questions of contemporary particle
physics: What is the nature of the Higgs boson? Can we successfully describe
the interaction of fundamental particles at high energies? Is there physics
beyond the Standard Model at the LHC? The capability to predict and describe
such observables at next-to-next-to-next-to-leading order (N$^3$LO) in QCD
perturbation theory is paramount to fully exploit these experimental
measurements. We describe the current status of N$^3$LO predictions and
highlight their importance in the upcoming precision phase of the LHC.
Furthermore, we identify key conceptual and mathematical developments necessary
to see wide-spread N$^3$LO phenomenology come to fruition.
