Electrical Impedance Tomography (ElT) is an emerging imaging technique with applications in the medical field and in the field of industrial process tomography (lPT). Until recently, data acquisition and image reconstruction schemes have been constructed with the assumption that the object being imaged is two-dimensional. In recent years, some research groups have started to address the third dimensional aspects of ElT by both building three dimensional enabled data acquisition systems and solving the three dimensional Forward Problem numerically since this allows the possibility of modelling complex shapes. However, solving the Forward Problem analytically is still very attractive as an analytical solution does not depend on the way the domain has been meshed. Furthermore, if dynamic images are reconstructed which are less sensitive to the model of the electrodes employed, the shape of the object being imaged and the position of the electrodes, an analytical solution to the Forward Problem can be used to reconstruct dynamic three dimensional images. This thesis will start by describing how a full analytical solution for a finite right circular cylinder (which approximately models the human thorax) on which two electrodes have been placed, is derived. It will be shown that the analytical solution has two different forms. Results will be presented detailing the convergence performance of the two different forms as well as comparisons between the analytical solution and experimentally obtained data. Finally three dimensional images reconstructed using these methods will be presented. In order to better approximate the shape of the human thorax, the above work has been extended to provide an analytical solution for an elliptical cylinder and this is presented in this thesis for the first time together with some simulation results. Today in Multi-frequency Electrical Impedance Tomography (MEIT), new hardware for recording measurements operating above 1MHz is now available. This high operating frequency raises the question of the validity of the employed quasi-static conditions used in the associated Forward Problem modelling. It is important to be able to determine when the quasi-static conditions fail and to investigate the differences between a solution to the Forward Problem based on quasi-static conditions and the one based on non quasistatic conditions at these frequencies. This thesis details the derivation of a new analytical solution based on non quasi-static conditions for a finite right circular cylinder having two electrodes placed on its boundary. Some comparisons between the new analytical solution and data obtained from in-vitro experiments will be presented in this thesis. A comparison between the new analytical solution and the analytical solution derived earlier in this thesis (which is based on quasi-static conditions) is also conducted. Whilst these results are preliminary results, they reveal that for situations associated with imaging the human thorax the quasi-static assumption appear violated when most modern MEIT systems are employed. This frequency dependent three dimensional analytical Forward Problem work has wide ranging implications for the future of MEIT. The thesis will conclude with some initial thoughts on how to incorporate anisotropy into three dimensional Forward Problem solutions.