We review recent studies, experimental and theoretical, of the nonlinear magnetic response of Mn₁₂ single-molecule magnets. Compared with the linear ac susceptibility, which has become the standard tool to ascertain whether a molecular cluster possesses magnetic memory (i.e. whether it behaves as a “single-molecule magnet”), the nonlinear susceptibility provides additional information on the relaxation process at a little cost in terms of experimental time and complexity. The nonlinear dynamic susceptibility depends not only on the relaxation times, as the linear susceptibility does, but also on how sensitive the relaxation process is to external magnetic fields. We show that the presence of spin quantum tunneling, and its strong dependence on external bias that detune the tunneling levels, gives rise to a very large contribution to the nonlinear response of Mn₁₂ clusters. Just like tunneling itself, this “quantum nonlinearity” can be “switched off and on” by external magnetic fields. By studying the orientational dependence of the nonlinear susceptibility, we estimate a bound for the decoherence time due to the coupling to the phonon bath. We find that, for tunneling via thermally activated states of Mn₁₂ acetate, decoherence is not limited by the lifetime of the excited states, but by a much shorter timescale of order 10⁻¹¹s.