Specialised experimental methods have been developed to study high rovibrationally excited molecular ions. These methods, based on an ion beam experiment, include enabling microwave or millimetre wave radiation to interact with the ion beam, and an indirect detection technique which relies on electric field-induced dissociation. Passage of the ion beam through an electric field lens results in selective fragmentation of high-lying energy levels; the ensuing fragment ions are separated from all other ions and detected. Spectroscopic transitions either to or from a level undergoing field-induced dissociation can be detected as a change in this fragment ion current. These methods have enabled an investigation of the simplest molecular ion, H₂⁺. Four electronic transitions between high-lying rovibrational levels in the 1sσ_g ground electronic state, and rovibrational levels supported by the long-range minimum of the 2pσ_u first excited state, have been observed. These include an improved recording of a previously detected microwave transition, and three new millimetre wave transitions. Unexpected splittings were observed in three of these spectra, which have been attributed to an electronic g/u symmetry breaking effect caused by the Fermi contact interaction. The long-range interaction within the He�Ar⁺ molecular ion has also been investigated. The development of further experimental methods, including 2-photon techniques and Zeeman spectroscopy, was necessary to interpret the spectrum. A total of 68 microwave and millimetre wave transitions were observed, of which 66 have been assigned. The pattern of 37 rovibronic energy levels lying within 7.5 cm⁻¹ of the lowest dissociation limit, He(¹S₀) + Ar⁺(²P₃/₂), has been determined.