Massless collisionless matter is described in general relativity by the massless Einstein--Vlasov system.  I will present key steps in a proof that, for asymptotically flat Cauchy data for this system, sufficiently close to that of the trivial solution, Minkowski space, the resulting maximal development of the data exists globally in time and asymptotically decays appropriately.  By appealing to the corresponding result for the vacuum Einstein equations, a monumental result first obtained by Christodoulou--Klainerman in the early '90s, the proof reduces to a semi-global problem.  A key step is to gain a priori control over certain Jacobi fields on the mass shell, a submanifold of the tangent bundle of the spacetime endowed with the Sasaki metric.