  We study the constraints imposed by perturbative unitarity on the new physics
interpretation of the muon $g-2$ anomaly. Within a Standard Model Effective
Field Theory (SMEFT) approach, we find that scattering amplitudes sourced by
effective operators saturate perturbative unitarity at about 1 PeV. This
corresponds to the highest energy scale that needs to be probed in order to
resolve the new physics origin of the muon $g-2$ anomaly. On the other hand,
simplified models (e.g.~scalar-fermion Yukawa theories) in which renormalizable
couplings are pushed to the boundary of perturbativity still imply new on-shell
states below 200 TeV. We finally suggest that the highest new physics scale
responsible for the anomalous effect can be reached in non-renormalizable
models at the PeV scale.
