A study of the Compton Scattering Camera in application to single photon emission computed tomography has been carried out and is presented in this thesis. It is shown that conventional gamma camera technology is fundamentally restricted by the collimator. The collimator places restrictions on camera sensitivity and on achievable reconstructed spatial resolution. The Compton scattering camera is proposed as an alternative to conventional gamma camera technology. The Compton scattering camera uses the process of Compton scattering to determine where a photon has come from. This results in a significant improvement to camera sensitivity and opens up the potential for improved spatial resolving power. An analytical model describing the Compton scattering camera is applied and studied. Suggestions are made regarding the design of a practical camera. It is shown that while a significant gain in sensitivity is achieved by the use of the alternative camera, many more measurement data bins are generated. The latter leads to poor photon counts in each data bin in situations encountered in practical medical imaging. Suggestions are made as to how to combat this problem. The use of Compton scattering to localise a photon direction vector leads to a new and complicated image reconstruction problem. Iterative algorithms have been devised for the image reconstruction by others and are reviewed herein. Progress towards developing a theory of direct image reconstruction is reported. To this end the notion of the cone-surface projection is introduced. It is shown that a certain subset of data, namely the restricted cone-surface projection, is invertible leading to a formula giving the source distribution in terms of the projections. Hence, the reconstruction problem, in the absence of noise uncertainties, is overspecified. Two possible reconstruction paths are outlined. One is direct reconstruction based on the restricted cone-surface projections. The other forms the cone-beam projections from the cone-surface projections and employs standard cone-beam reconstruction. Results of computer simulation of the reconstruction paths are reported. The direct reconstruction is robust with respect to some angular uncertainty and missing low angle scattering. The formation of the parallel-ray projection from the restricted cone-surface projection appears to be adversely susceptible to such measurement uncertainty. These studies are carried out without accounting for photon noise. For the algorithms to be practically useful, it is necessary that the theorems relating to the restricted cone-surface projection be generalised to cover the full cone-surface projection. Suggestions as to how this may be done are given. To perform the direct reconstruction from the restricted cone-surface projection. it is necessary to numerically evaluate the Hankel transform. The literature on Hankel transform algorithms is somewhat fragmented; a review of Hankel transform algorithms is therefore presented. Results of testing of a variety of zero-order Hankel transform algorithms are reported. It is found that algorithms based on the trapezoidal rule and the back-projection method with Fourier interpolation lead to the most accurate Hankel transforms in general. Unfortunately these algorithms are relatively computationally expensive. More efficient algorithms, such as the projection-slice method with Hansen and Law's method of evaluating the Abel transform, may be suitable in applications where less accuracy can be tolerated.