Future generations of gravitational wave detectors will require significant progress in the reduction of all forms of noise affecting the system. One form of noise it is critical to reduce is thermal noise, which can be described as the consequence of the atoms which make up the measurement optics experiencing vibrations because of their non-zero temperature. The dielectric multilayer coatings of the mirror in an interferometric gravitational wave detector are known to contribute significantly to the overall levels of thermal noise. The next generation of gravitational wave detectors may need to use exotic coatings, cryogenic operating temperatures and silicon mirror substrates in an effort to mitigate the effects of thermal noise. Chapter Two describes thermal noise in detail, and introduces the concepts of substrate noise, coating noise, thermoelastic dissipation, mechanical loss and the formulae used to calculate them. Chapter Three describes the current state of research on the factors affecting mechanical loss in dielectric coatings. The technique of probing the structure and dissipation characteristics of materials by assessing the shape and position of the low temperature excess loss feature known as a Debye peak is introduced. The cryogenic mechanical loss measurement apparatus used in Chapters Four, Five and Six is described and characterised. Chapter Four concerns the variation of mechanical loss of ion-beam sputtered silica coatings with temperature and investigates the effects of heat-treatment upon them. The low-temperature Debye peak was found in some modes of a sample heat treated at 300oC and an Arrhenius analysis provided a characteristic energy for the dissipation process of (17.3 ± 2.3)meV. Further heat treatment of silica at 600oC and 800oC appears to narrow the Debye peak, which is thought to be indicative of the narrowing of the distribution of bond angles in the amorphous silica network. Hafnia is investigated as an alternative coating material in Chapter Five. The mechanical loss of hafnia heat-treated at 300oC was measured and two excess loss features were discovered, one below 100K and one above 200K. Electron scattering measurements indicate that this sample may already have developed polycrystalline regions which are known to be connected to high levels of mechanical loss. The mechanical loss of an un-heat-treated hafnia coating is also measured and an extremely low coating loss of 1.87 × 10−5 is found at 20K. Chapter Six describes an experiment to find the mechanical loss of a hydroxycatalysis bond between silicon cantilevers at temperatures between 10K and 300K.This new technique for the measurement of the mechanical loss of bond material produced a minimum upper limit of the bond loss of (0.13 ± 0.03) occurring in the fundamental mode at 80K and upper limit of the bond loss of (0.19 ± 0.07) occurring in the third mode at 15K. Chapter Seven describes the development and testing of a nodal support system to enable cryogenic measurements of cylindrical bulk mirror substrates to be made. The efficacy of the support varied significantly with the frequency of the mode and the cryogenic measurements were partially successful. The major results in this work are the successful measurements of the mechanical loss of amorphous hafnia coatings at low temperatures and the use of a structure made from hydroxy-catalysis bonded silicon cantilevers to obtain an upper limit for the mechanical loss of the bond material. These results may inform technological advances that reduce the level of thermal noise experienced in future gravitational wave detectors.