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Abstract: Inelastic scattering from $N$ coherent spins, or any two-state system, scales as $N^2$, invalidating the Born approximation at large $N$ even at weak coupling. We develop a nonperturbative theory of this superradiant scattering and its backreaction on the spins. Since one particle can exchange many quanta, the target's energy-transfer and energy-variance rates keep their favorable $N^2$ scaling even after the cross section saturates, unless a conservation law limits each encounter to exchange of one quantum. We apply the theory to millicharged-particle searches, as well as neutron and charged-particle beams that can test quantum protocols for detecting cosmic relics.