Electrical impedance tomography (EIT) is a non-intrusive and portable imaging technique which has been used widely in many medical, geological and industrial applications for imaging the interior electrical conductivity distribution within a region from the knowledge of the injected currents through attached electrodes and resulting voltages, or boundary potential and current flux. If the quantities involved are all real then EIT is called electrical resistance tomography (ERT). The work in this thesis focuses on solving inverse geometric problems in ERT where we seek detecting the size, the shape and the location of inner objects within a given bounded domain. These ERT problems are governed by Laplace’s equation subject either to the most practical and general boundary conditions, forming the socalled complete-electrode model (CEM), in two dimensions or to the more idealised boundary conditions in three-dimensions called the continuous model. Firstly, the method of the fundamental solutions (MFS) is applied to solve the forward problem of the two-dimensional complete-electrode model of ERT in simplyconnected and multiple-connected domains (rigid inclusion, cavity and composite bimaterial), as well as providing the corresponding MFS solutions for the three-dimensional continuous model. Secondly, a Bayesian approach and the Markov Chain Monte Carlo (MCMC) estimation technique are employed in combinations with the numerical MFS direct solver in order to obtain the inverse solution. The MCMC algorithm is not only used for reconstruction, but it also deals with uncertainty assessment issues. The reliability and accuracy of a fitted object is investigated through some meaningful statistical aspects such as the object boundary histogram and object boundary credible intervals.