This thesis aims at introducing the reader to the mathematical concepts behind the imaging technique X-ray tomography, commonly known as a CT scan. It includes the derivation of the filtered and unfiltered backprojection reconstruction methods for noise-free data. It was concluded that for noisy data, X-ray tomography is an ill-posed, and therefore unstable, inverse problem that needs regularization in order to produce adequate reconstructions. The methods of truncated singular value decomposition regularization and Tikhonov regularization are derived, analyzed and compared using simulated as well as real-life data. It was found that both methods can produce stable reconstructions, but that Tikhonov regularization is less sensitive to parameter choice.