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In this thesis we present new ideas and mathematical insights in the field of Compton Scattering Tomography (CST), an X-ray and gamma ray imaging technique which uses Compton scattered data to reconstruct an electron density of the target. This is an area not considered extensively in the literature, with only two dimensional gamma ray (monochromatic source) CST problems being analysed thus far. The analytic treatment of the polychromatic source case is left untouched and while there are three dimensional acquisition geometries in CST which consider the reconstruction of gamma ray source intensities, an explicit three dimensional electron density reconstruction from Compton scatter data is yet to be obtained. Noting this gap in the literature, we aim to make new and significant advancements in CST, in particular in answering the questions of the three dimensional density reconstruction and polychromatic source problem. Specifically we provide novel and conclusive results on the stability and uniqueness properties of two and three dimensional inverse problems in CST through an analysis of a disc transform and a generalized spindle torus transform. In the final chapter of the thesis we give a novel analysis of the stability of a spindle torus transform from a microlocal perspective. The practical application of our inversion methods to fields in X-ray and gamma ray imaging are also assessed through simulation work.