This article reviews the discovery, exploration, and application of negative-ion resonances in inelastic electron scattering by molecules adsorbed on surfaces. A major theme of the review is the degree to which the properties of resonances in free molecules are perturbed by adsorption. The influence of the surface upon the energy, lifetime (width), symmetry, and decay channels of molecular resonances is discussed, in the light of both experimental and theoretical studies of a wide range (from diatomic molecules to polymers) of both weakly bound (physisorbed) and strongly bound (chemisorbed) molecules. The metallic image potential, electron scattering by the atoms of the surface, and chemical bonding in chemisorption systems are found to be key factors in determining the energy, width, and symmetry of resonances in molecular adsorbates. In the case of oriented adsorbed molecules, the angular distribution of scattered electrons is found to reflect not only the symmetry of the resonant state (as in the gas phase), but also the orientation of the molecular axis. Coherent elastic electron scattering by the surface can modulate the angular distributions, as well as the shape of the resonance profile. Selection rules that govern the observed resonance behavior are discussed. A further consequence of adsorption is the enrichment of the range of channels into which resonances may decay, and the excitation of both molecule-surface and intermolecular vibrational modes has been established. The article concludes with an evaluation of future prospects for the investigation and application of resonances in adsorbed molecules.