Positron Emission Tomography (PET) is a pillar of modern diagnostic imaging, allowing non-invasive, sensitive and specific detection of functional changes in several disease types. In endocrinology, the precise localisation of small functioning tumours of the pituitary or adrenal glands is crucial for planning curative surgery or radiotherapy. While PET imaging shows good promise for this task, initial studies suggest significant room for improvement, with improved PET imaging and subsequent more accurate localisation opening up the possibility for more adapted therapies. In dementia, the accurate quantification of PET images is key for the early detection of disease. Improved PET imaging may allow for earlier detection of dementia while asymptomatic and increased sensitivity to assess and monitor treatment once appropriate drugs have been found. In this project mathematicians team up with researchers and clinicians from Addenbrooke's Hospital Cambridge, Dementias Platform UK (DPUK), GE Healthcare and University College London (UCL) for improved diagnosis and localization for tumours in endocrinology and earlier diagnosis of dementia with improved PET imaging. In particular, we investigate modern PET reconstruction approaches based on advanced mathematical methods to increase the PET image resolution and contrast, while keeping computational complexity low, thereby directly benefiting clinical workflow.