Exact, analytic expressions have recently been derived for the magnetisation of thin, type-II, superconducting films in the critical state. They predict significant differences in the physical behaviour as compared with the well-known expressions obtained by Bean for infinite plates and cylinders, and this includes the AC susceptibility. We review the experimental evidence available from studies on thin films, and find χ′ versus χ″ results in very good agreement with the new theories, but the predicted dependence on the amplitude of the modulation field is not convincingly shown for the fundamental, and not at all for the harmonics.