Understanding and controlling the behaviour of metal-semiconductor contacts is central to the operation of modern electronics. Schottky contacts, formed as a result of a potential barrier at the metal-semiconductor interface, substantially inhibit current flow in the reverse direction. The advent of thin-film electronics and the increasing use of disordered materials, such as oxide semiconductors, have complicated the relationship between the Schottky barrier and the current, making further investigation necessary. By fabricating thin-film Schottky diodes using the oxide semiconductor indium gallium zinc oxide, a remarkably strong dependence of the reverse current upon semiconductor thickness is demonstrated. With the aid of device simulations, the dependence is attributed to a spatially inhomogeneous Schottky barrier height. To add further depth to the current understanding of thin-film Schottky diodes, an analytical model is devised which also incorporates the behaviour of barrier inhomogeneities. As the mechanism for transport across the barrier is material dependent, theories are developed for different regimes. Building upon the work on thin-film Schottky diodes, Schottky source transistors, which are a constructive combination of a Schottky diode and a thin-film transistor, are studied. By developing a methodology to manipulate the shape of the Schottky barrier in these Schottky source transistors, this work demonstrates an extremely high voltage-amplification gain of 29,000, which is orders of magnitude higher than a conventional Si transistor. These same devices demonstrate almost total immunity to negative bias illumination temperature stress, the foremost bottleneck to using oxide semiconductors in major applications such as display drivers. Furthermore, transistors fabricated with a 360 nm channel length display no obvious short-channel effects, another critical factor for high-density integrated circuits and display applications. Finally, although the channel material of conventional transistors must be a semiconductor, by demonstrating a high performance transistor with a semi-metal-like indium tin oxide channel, the range and versatility of materials has been significantly broadened.