The STM's ability to image adsorbates depends on the probability that the electronic states localized on the adsorbed molecule contribute to the tunnelling current. When the STM images are dominated by the substrate, any interpretation in terms of the actual positions of the atoms within the molecule is controversial. A criterion is presented for deciding whether an STM observation is of the adsorbed molecule directly or of the indirect effect of the molecule on the current from the substrate. It is based on analysis of the wave function associated with the tunnel current obtained by a self-consistent molecular orbital calculation, within the LCAO and cluster framework at CNDO level. The results for a range of molecules and polymers are in this case fully consistent with the criterion. The data suggest that any credible high-resolution STM observation of adsorbed species requires a contribution to the wave function not less than 50%. The notorious observed differences in contrast of co-adsorbed species are mainly attributed to consideration differences among individual contributions to the tunnel current.