In this thesis a recently developed energy resolving x-ray detector (Medipix) is used to investigate potential medical applications of spectral x-ray imaging. Computed Tomography (CT) is one of the most important medical imaging modalities. Recent developments in CT techniques include dual-energy CT, where images are taken with two different x-ray spectra by either using two x-ray tubes operated at different voltages, or modulating the operating voltage of a single tube. These techniques provide spectral information in the CT dataset but are limited to what can be achieved by manipulating the x-ray source, since the detectors used in current CT systems are unable to provide spectral information about the detected x-rays. A preliminary investigation of the use of the Medipix detectors for two different medical applications is presented. The first, applications is imaging of blood vessels for diagnosis of vascular diseases, and the second, characterising and measuring the energy dependence of x-ray attenuation in fat and liver tissue using the Medipix2 detector. This second investigation is part of work towards (eventually) quantifying the fat content of liver tissue in vivo, which is important for the early diagnosis of fatty liver disease. While an early attempt to identify iron fluorescence x-rays in a Monte-Carlo simulation of blood vessel x-ray image was not successful, the potential for improving image contrast using the changes in x-ray attenuation at the iodine k-edge iodine have been investigated in a series of further simulations and appears to be feasible. The potential use of spectral imaging to differentiate and quantify tissues without the need for added contrast material has been investigated by using a Medipix2 detector to measure the energy dependence of x-ray absorption in fat and liver tissue. The results of this experimental work show significant differences in x-ray attenuation between these two tissues that suggest this form of spectral imaging may be useful in practice.