This research is focused on the numerical solution of the inverse conductivity problem, widely known as electrical impedance tomography (EIT). The EIT problem is concerned with imaging electrical properties, such as conductivity (sigma) and permittivity (epsilon), in the interior of a body given measurements of d.c. or a.c. voltages and currents at the boundary. Given complete and perfect knowledge of the boundary data, the EIT problem is known to have a unique solution. In practice however, the data is noisy and incomplete. Hence, satisfactory solutions of the nonlinear ill-posed EIT problem are difficult to obtain. In this thesis, we introduce a family of variational formulations for the EIT problem which we show to have advantages over the popular output least squares approach. Output least squares seeks sigma and/or epsilon by minimizing the voltage misfit at the boundary measured in the L2 norm. In contrast, the variational methods ensure that the measured data is being fit in a more natural norm, which is not the L2 norm. These methods also introduce some natural regularization. Through extensive numerical experimentation, we compare the performance of our variational formulations with one another and with the standard least squares algorithm. Using the same data, we demonstrate that our variational algorithms produce better images without significantly increasing computational cost.