This paper proves that extremal black holes arise on the threshold of gravitational collapse. The authors construct smooth one-parameter families of spherically symmetric solutions to the Einstein–Maxwell–Vlasov system that interpolate between dispersion and collapse, with the critical solution being an extremal black hole. These solutions can be understood as beams of gravitationally self-interacting collisionless charged particles fired into Minkowski space. Depending on the parameter, the Vlasov matter either disperses or collapses. At the critical value, an extremal Reissner–Nordström black hole forms. No naked singularities occur at the extremal threshold. This phenomenon is called extremal critical collapse, and the paper presents the first rigorous result on the black hole formation threshold in general relativity. The paper also discusses the third law of black hole thermodynamics and event horizon jumping at extremality. It shows that in the Einstein–Maxwell–Vlasov model, a subextremal Reissner–Nordström apparent horizon can evolve into an extremal Reissner–Nordström event horizon in finite advanced time due to the incidence of charged Vlasov matter. The paper also presents conjectures about the stability of extremal critical collapse and the behavior of extremal black holes in various models, including the Einstein–Maxwell-charged scalar field model and the Einstein vacuum equations. The results demonstrate that extremal critical collapse is a new phenomenon in general relativity, with implications for the understanding of black hole formation and thermodynamics.This paper proves that extremal black holes arise on the threshold of gravitational collapse. The authors construct smooth one-parameter families of spherically symmetric solutions to the Einstein–Maxwell–Vlasov system that interpolate between dispersion and collapse, with the critical solution being an extremal black hole. These solutions can be understood as beams of gravitationally self-interacting collisionless charged particles fired into Minkowski space. Depending on the parameter, the Vlasov matter either disperses or collapses. At the critical value, an extremal Reissner–Nordström black hole forms. No naked singularities occur at the extremal threshold. This phenomenon is called extremal critical collapse, and the paper presents the first rigorous result on the black hole formation threshold in general relativity. The paper also discusses the third law of black hole thermodynamics and event horizon jumping at extremality. It shows that in the Einstein–Maxwell–Vlasov model, a subextremal Reissner–Nordström apparent horizon can evolve into an extremal Reissner–Nordström event horizon in finite advanced time due to the incidence of charged Vlasov matter. The paper also presents conjectures about the stability of extremal critical collapse and the behavior of extremal black holes in various models, including the Einstein–Maxwell-charged scalar field model and the Einstein vacuum equations. The results demonstrate that extremal critical collapse is a new phenomenon in general relativity, with implications for the understanding of black hole formation and thermodynamics.