In conventional experiments, diffusion was treated as a scalar quantity, based on models assuming spherical or isotropic diffusion. In most living tissues, this is an oversimplification as diffusion is not isotropic and must be treated as a tensor quantity. As such, the likely errors arising from performing diffusion measurements assuming spherical or cylindrical symmetry were investigated using computer simulations. Large and unpredictable errors were found to result if diffusion was not treated as a tensor quantity and models assuming spherical or cylindrical symmetry were used. Diffusion measurements are also highly susceptible to the effects of experimental noise. Previously suggested measurements as to the extent of noise corruption in diffusion experiments are generally found to be extremely dependent on the experimental parameters used and so are difficult to apply in a general situation. To overcome this, a measurement of noise was found that gives an analysis that is largely independent of the experimental conditions. Achieving an accurate quantitative analysis has been shown to require a careful balance between obtaining a high enough degree of diffusion weighting, while achieving sufficient signal-to-noise ratio. A theoretical method was developed for producing reliable diffusion measurements by optimising the diffusion weighting b-value and the number of acquisitions obtained. Both in vitro and in vivo data were found to give reliable quantitative data from an acquisition scheme based on the theoretical method. Many different ways for displaying diffusion data have been proposed. An analysis of the different levels of contrast and sensitivity arising from various diffusion anisotropy indices was also undertaken, resulting in the development of a method for displaying diffusion data with improved contrast. During the course of this work quantitative diffusion imaging has been performed in a clinical setting on over 100 acute stroke patients, 16 head injury patients and 5 CJD patients.