Increasingly in medical imaging has emerged an issue surrounding the reconstruction of noisy images from raw measurement data. Where the forward problem is the generation of raw measurement data from a ground truth image, the inverse problem is the reconstruction of those images from the measurement data. In most cases with medical imaging, classical inverse Radon transforms, such as an inverse Fourier transform for MRI, work well for recovering clean images from the measured data. Unfortunately in the case of X-Ray CT, where undersampled data is very common, more than this is needed to resolve faithful and usable images. In this paper, we explore the history of classical methods for solving the inverse problem for X-Ray CT, followed by an analysis of the state of the art methods that utilize supervised deep learning. Finally, we will provide some possible avenues for research in the future.