With the latest developments in magnetic resonance imaging (MRI) technology, particularly in the areas of high-field superconducting magnets, high-performance ultra-short gradient coils and high radio-frequency (RF) excitation devices; the interaction of electromagnetic fields generated by the new generation of imagers and patients, healthcare workers as well as system components has recently attracted substantial attention. Due to the complexity of the electromagnetic field - tissue and field - metal interactions, computational modelling plays an essential role in the analysis, design and development of modern MRI systems. Recent progress in the development of MRI superconducting magnets has resulted in a considerable increase in human exposure to very large static magnetic fields of up to several Tesla. Body movement through these fields can cause the induction of currents that are potentially above the regulatory limits. In addition to that, novel imaging sequences demand very large magnitudes and high switching rates in magnetic field gradients, which are known to be the prime source of frequently reported peripheral nerve stimulation (PNS) sites in the patients. When highfrequency fields are employed to excite a spin ensemble during MRI imaging, electromagnetic energy is coupled with the tissue and deposited within, which causes regional temperature elevations within the patient, thus leading to possible tissue/cell injury. Overall, electromagnetic field – tissue interaction is a hot topic of research and requires further analysis and consideration. Apart from interacting with the patient, electromagnetic fields produced by the imager also couple to the conducting materials in the MRI system to induce eddy currents that degrade image quality. The eddy current manifestations are a significant concern in MRI and require accurate prediction models, analysis schemes and control methods. Overall this thesis is concerned with computational bioelectromagnetics and associated effects such as concomitant thermal changes. The developed methods are also used in novel design scenarios. In part, this research engages the numerical computational modelling of patient and healthcare worker exposures to strong static and low-frequency pulsed magnetic fields produced by different main superconducting magnets and gradient coils respectively. The main focus herein is on the computation of electric field and current density distributions and levels within tissue-equivalent models of males and females. Various exposure scenarios and setups are considered in the work to evaluate, analyze, compare, comprehend and predict the worst-case field induction in the tissue. This information is particularly useful in terms of compliant activity around and within the clinical MRI imagers. The thesis also details the development and utilization of modified finite-difference time-domain (FDTD) methods in cylindrical space for numerical modelling of lowfrequency transient eddy currents induced within realistic cryostat vessels during both longitudinal and transverse magnetic field gradient switching. In addition, transient eddy currents are numerically evaluated using the method and incorporated into a longitudinal gradient coil design process. In the optimization procedure the gradient coil is modified so that the fields created by the coil and the eddy currents combine together to generate spatially homogeneous gradients that follows the desired temporal variation. In that way the eddy currents are neither prevented nor minimized but rather constructively used in shaping uniform space-time magnetic field gradients. Furthermore, the research presents linear and non-linear heat transfer computational models on the basis of the conventional Penne’s bio-heat transfer equation. The nonlinear model is verified against experimental temperature results from a hyperthermia study on a mouse using a 150 kHz induction coil, while the linear model is used directly in a study on rats under the exposure of high-frequency volume resonators (0.5 - 1 GHz). The thermal models find applications in modelling the deposition of electromagnetic field energy within tissue and computation of associated thermal effects in high-field MRI.