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ABSTRACT: It is argued that radiative decays of scalars a0/f0(980) can serve as a decisive tool in establishing the nature of the latter. In particular, predictions for the widths of the radiative decays S->gamma V (S=a0/f0(980), V=omega/rho/gamma) are given in the framework of the molecule model of the scalars. Finite-range corrections are discussed in detail for the two-gamma decays of hadronic molecules, with a special attention payed to the interplay of various scales involved in the problem and to the gauge invariance of the amplitude. The results are applied to the two-photon decay of the f0(980), and the existing experimental data on this decay are argued to support the molecule assignment for the scalar f0(980).