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This dissertation describes work carried out between June 1987 and October 1991, in the Low Temperature Physics Group at the Cavendish Laboratory, Cambridge. The aim of this work was to use electron tunnelling spectroscopy to measure the density of excitation states of the recently discovered high-temperature superconductors. Tunnelling is the most sensitive method for measuring a superconductor’s energy gap, and historically has provided important evidence for the microscopic mechanism of superconductivity in conventional metals. It was hoped that electron tunnelling would prove equally successful in revealing the mechanism of superconductivity in these new materials. Preliminary experiments showed that a thick, degraded surface layer prevented preparation of high-quality tunnel junctions by conventional evaporation techniques. For this reason, apparatus for the formation and fine control of low-temperature point-contact junctions was constructed, together with new measurement electronics and a computer-controlled data-acquisition system. To test this apparatus, point-contact junctions were formed on conventional superconductors. By increasing pressure of the tip on the sample the junction could be moved from the tunnelling to the metallic regime. Point-contact measurements were then made on a number of ceramic, single-crystal and thin-film high-temperature superconducting materials; some not previously investigated by tunnelling. Although ‘gap-like’ structure was occasionally observed, anomalous features (e.g., voltage-dependent background, broadening, large zero-bias conductance) were always present and usually dominated the tunnelling characteristics. These complicate estimation of the energy gap and preclude measurement of more subtle properties such as gap anisotropy or the effective phonon spectrum, α2F. The origins of these non-ideal features have been much debated in the literature and are reviewed in this dissertation. In the case of thin films deposited by laser ablation the tunnelling characteristics were dominated by single-electron tunnelling effects (Coulomb gap and staircase structure). The results suggest that the surface region consists of numerous, isolated normal metal particles.