By sensitising the magnetic resonance signal to the diffusion of water molecules in tissue, diffusion-weighted magnetic resonance imaging provides a means of assessing tumour microstructure non-invasively. Such measurements have the potential to provide important information about tumour development and the response of tumours to treatment, but the way in which different tissue properties affect the diffusion-weighted signal remains unclear. Through simulations, in vivo studies and phantom experiments, this thesis investigates the relationship between the diffusion-weighted signal, the pulse sequence parameters used for acquisition, and microstructural properties of tumours.The use of oscillating gradient pulse sequences on a clinical scanner was investigated initially, with theoretical and practical considerations leading subsequent work to focus on pulsed gradient sequences. The forward problem of predicting the diffusion-weighted signal for given combinations of tissue properties and sequence parameters was addressed numerically through Monte Carlo simulations, focussing on how tumour cell size, intracellular volume fraction and membrane permeability affect the signal. These simulations allowed the sensitivity of the signal to changes in these tissue properties to be investigated, revealing how sensitivity depends on sequence parameters as well as the specific microstructural configuration. By repeating the simulations using the specific sequence parameters used in a clinical and preclinical study, the sensitivity of the implemented protocols was assessed, and linked to the experimental findings. The preclinical study illustrated the importance of the diffusion time in determining the sensitivity to treatment-induced changes in tumours, with larger post-treatment signal changes observed at longer diffusion times. These trends were qualitatively reflected in the sensitivity analysis derived from the simulations. Finally, the inverse problem of estimating microstructural properties from the diffusion-weighted signal was addressed using a physical phantom designed as a simple mimic of tumour tissue. By fitting a biophysical model to the diffusion data, the size and volume fraction of the approximately spherical 'cells' were estimated. The radius was slightly underestimated compared with that determined from independent measurements, the fitted volume fraction was plausible, and parameters were found to be estimated with reasonably good precision.