Medical imaging is a cornerstone of modern healthcare. The ability to acquire images from inside a patient has revolutionised the way doctors diagnose and treat diseases, with almost all clinical pipelines now involving imaging to some degree. The development of these imaging methods has led to the field of medical image computing, where a multitude of tools and techniques have been proposed to aid clinicians and researchers in interpreting and analysing these images. One such family of techniques involves generating synthetic medical images. Image synthesis techniques are wide and varied, ranging from basic phantoms, to disease atlases, to high resolution photo-realistic subject-specific images. Their applications are similarly diverse: for developing novel acquisition protocols, training and testing algorithms, visualising changes in disease, predicting particular image types from others, and improving image quality and resolution. This thesis examines the use of medical image synthesis with a particular focus on applications in neurodegenerative diseases. A method for synthesising subject-specific non-pathological images from pathological images is first proposed and used for the unsupervised brain lesion segmentation. We next show how generative adversarial networks (GANs) can be used to both analyse the structural changes seen in patients with Alzheimer's disease, and to add or remove these changes from patient images to produce a subject-specific prediction of disease progression. Finally we investigate how GAN-derived synthetic data can be used to increase the size of training datasets, and under what conditions this additional data can lead to an improvement across a variety of segmentation tasks. Within this context we explore two situations: where only a small amount of labelled data is available, and where a large amount of unlabelled data is also available. We show that the proposed methods can lead to significant improvements in segmentation results, especially when a small amount of labelled data is available.