The aim of this area of research is to provide techniques which would allow long-term low temperature storage and subsequent transplantation of human organs, such as the kidney. A major difficulty in organ cryopreservation is that of obtaining rapid warming of the frozen tissue, whilst maintaining a uniform temperature profile . Before cooling, the organ is perfused with a 'cryoprotectant', a solution which reduces freezing injury. Warming rates that can be achieved by thermal conduction cannot exceed about 1K/min without damaging the exterior of the organ. Experimental evidence from a range of cell and organ systems suggest that warming should be much more rapid. One proposed warming technique employs dielectric heating using electromagnetic energy at microwave frequencies. Crucial to the design of any such rapid warming system are the electrical properties of the material to be heated. This dissertation describes two measurement systems that have been developed for measuring the electrical properties of cryoprotectants and of perfused rabbit kidney tissue. Dielectric measurements were made with both time and frequency domain instrumentation, over a frequency range of 50MHz to 2.6GHz, and a temperature range of -30'C to +20'c. The more successful of the two systems uses an open-ended coaxial probe with a new, simplified, error correction and calibration technique. The measurements show that electrical properties of perfused tissue are heavily influenced by those of the cryoprotectant in this frequency range. A new low conductivity cryoprotectant has been developed which has more favourable electrical properties. There is a strong suggestion that using such a cryoprotectant, rapid warming may become a practical proposition around 430MHz.