  This article is devoted to the energy critical hyperbolic Yang--Mills system
in the $(4+1)$ dimensional Minkowski space, which is considered by the authors
in a sequence of four papers. The final outcome of these papers is twofold: (i)
the Threshold Theorem, which asserts that global well-posedness and scattering
hold for all topologically trivial initial data with energy below twice the
ground state energy, and (ii) the Dichotomy Theorem, which for larger data in
arbitrary topological classes provides a choice of two outcomes, either a
global, scattering solution or a soliton bubbling off. In the last case, the
bubbling off phenomena can happen either (a) in finite time, triggering a
finite time blow-up, or (b) in infinite time. Our goal here is to describe
these results, and to provide an overview of the flow of ideas within their
proofs.
