This project aims to develop computational imaging algorithms for 3D image reconstruction from spectral X-ray CT (Spectral Photon Counting Computed Tomography) measurements. This new modality finds a wide range of applications and has significant advantages over its predecessor, and current state of the art: the scalar X-ray CT. Apart from an enhanced spatial resolution, the anatomical information it provides includes a characterisation of the chemical composition of the materials/tissues within the imaged domain, thus enabling simultaneous image reconstruction and domain classification. The key scientific question pertinent to this problem is how to best exploit the spectral resolution in the data to further enhance the information content of the reconstructed images. Moreover, the richness of spectral information comes at a cost of higher computational complexity, which requires new algorithms that can achieve high quantitative imaging performance in short times and moderate resources. Recently, Deep Learning was shown to provide high-quality reconstruction performance in scalar X-ray CT while circumventing some of the computational challenges. To achieve such a performance in spectral X-ray CT we propose a two-step approach where we first learn which materials are present in the domain by solving a spectral unmixing problem, and then use the unmixed (material-specific) data to image the respective material(s) within the domain. To endow some degree of explainability, we will impart to the Deep Learning architectures some physics-inspired constraints, in the form of the spectral X-ray transmission (polyenergetic) models. The algorithms developed will be tested on biomedical data from calcified human valves that are readily available courtesy of our industrial collaborators, although other application areas can also be considered, e.g. homeland security, and material non-destructive testing. Familiarity with Pytorch/Tensorflow will be considered an advantage.