The electrical properties - permittivity and conductivity - of a material describe how electromagnetic waves behave in that material. Electrical properties are frequency-dependent parameters and, for a liquid sample, are measured with a dielectric probe and a network analyzer. This measurement technique is not feasible in vivo, but methods have been developed to make these measurements using magnetic resonance imaging (MRI). This work focuses on measuring conductivity, or the ability to conduct electric current. Mapping the electrical properties within the human body can provide important information for MRI safety and diagnostic applications. First, the specific absorption rate (SAR) in an MRI scan is proportional to conductivity, and limited to minimize the risk of heating in a subject. Knowledge of subject-specific conductivity maps could lead to better, subject-specific SAR estimation. Second, several small studies in recent years have shown that conductivity is elevated in malignant tumors as compared to healthy tissue. There are open research questions regarding the correlation between conductivity and other diagnostic metrics. Both of these applications benefit from accurate conductivity maps. In this work we describe three different methods for improving the accuracy of conductivity maps. The first is a novel regularized, model-based approach which we refer to as the Inverse Laplacian method. The Inverse Laplacian method resulted in lower reconstruction bias and error due to noise in simulations than the conventional filtering method. The Inverse Laplacian method also produced conductivity maps closer to the measured values in a phantom and with reduced noise in the human brain, as compared to the filtering method. The second is a method for combining multi-coil MRI data for conductivity mapping, because the use of multi-coil receivers can drastically improve the SNR in conductivity maps. The noise in the combined phase data using the proposed method was slightly elevated as compared to the optimal combination method, but the conductivity uniformity in a uniform gel phantom was greater than that of the optimal combination method. Furthermore, by visual inspection, the human brain conductivity calculated from data combined using the proposed method had minimal bias and noise amplification. Finally, we present a method for mapping conductivity tensors, as opposed to scalar values, which provides an additional layer of information to conductivity maps. Our proposed mathematical framework yields accurate tensor quantities provided the object can rotate 90 degrees in any direction. However, restricting the object rotation to mimic the constraints on a human subject yields slightly inaccurate results. We also present a dictionary-based approach to tensor calculations to try to improve the tensor estimates using restricted rotations.