A critical analysis of published thermal conductivity data is presented which highlights failures in data extrapolation, unexplained sudden drops in previously observed material data sets and the clarity of equipment design. Thermal conductivity measurements on a range of materials were performed using a Gifford-McMahon cryostat with the aim of consolidating current information on thermal properties as well as collecting data on new materials executed from 3 K to 30 K using the steady state method. The ability of the cryostat to accurately cycle between desired temperatures was verified and used for thermal stability tests of materials for the Mid-Infrared Instrument to be launched on the James Webb Space Telescope. Test undertaken on the electrical resistivity of aluminium foam were used to confirm the accuracy of the system. For steady state measurements of samples, the temperature across a known thickness of material was varied using a resistor heater of known power permitting the calculation of the thermal conductivity. The robustness of the system up to 20 K was shown using a Stainless Steel 316 sample, which agreed with expectations at the 2.1% level. The thermal conductivity of boron-doped silicon was measured, specifically looking at the conductivity across a hydroxide-catalysis bond. Such bonded silicon and silicate bonds are integral to the ongoing research for the next generation of gravitational wave detectors. The thermal conductivity of p-type boron-doped silicon was shown to increase form 24.06 - 46.44W/mK over the 4 -19 K temperature range. The improvement in the design features of cryogenic systems outlined are being implemented in labs at the University of Glasgow. The experimental setup developed during the course of this PhD project will also be used for data collection for medical diagnostic equipment.