The double electrostatic and cryogenic ion-beam storage ring, DESIREE, is a tool for studies of ions of atoms and molecules. It is unique in that it combines two important abilities, to store ions for very long times (up to hours) allowing the stored ions to relax to their lowest quantum states, and to study the interactions between oppositely charged ions stored in two separate storage rings with a common ion-beam merging section at low and well-controlled center-of-mass energies. We present results of experiments investigating the relaxation of molecular ions and of experiments studying the process of mutual neutralization between negative hydrogen ions and positive ions of different metals. A good understanding of the mutual-neutralization process is important for a correct modelling of stellar atmospheres and a correct quantitative determination of metal abundances from the strengths of the absorption features in stellar spectra.