This thesis concerns the design of gradient coils for magnetic resonance imaging systems. The method of design by genetic algorithm optimisation is applied to novel gradient geometries both by use of conventional computer facilities, and, by parallelisation of the design algorithm, on a supercomputer architecture. Geometries and regions of interests which are inaccessible to analytic solution are considered, and the criteria which are difficult to include in such algorithms, such as the robustness of the design, are also included. To exemplify this, in the first instance a two axis biplanar coil was designed and the performance of the genetic algorithm tested and evaluated. The coil was tested computationally; a working example was constructed and tested in a MRI scanner both on phantom objects and on a human knee. Consideration of the usefulness of the coil regions not optimised for linearity for image reconstruction was done. The gradient efficiencies of the final designs in the z and y directions respectively were 0.3 mTm-1A-1 and 0.4 mTm-1A-1 over a 15 cm diameter region of interest. The size of the interior of the gradient set was designed to be 40.0 cm x 24.4 cm x 40.0 cm, to fit within the confines of the bore of an existing scanner. The linearity in the primary direction over the region of optimisation was less than 5% for both coils. The algorithm was extended for operation on a Hitachi SR2201 supercomputer using parallelisation. The performance in this mode was evaluated and found to be favourable in comparison with the standard computer architecture, with an increase in speed in real time of a factor of-!llore than 40 in some configurations of the supercomputer. Various polygonal cross-section design shapes requiring the use of this improved computer performance were optimised and evaluated computationally. Such designs have previously been inaccessible to the genetic algorithm optimisation model. Tests were made between the performance of the genetic algorithm on various similar design problems, and simulated images from such gradient coils were produced. Finally an example of a transverse coaxial return path gradient coil is presented computationally. This coil had an internal diameter of 32 cm, a d external diameter of 44 cm and a length of 40 cm. It achieved a strength of 0.1 mTm- 1A -1 over a cylinder of diameter 20 cm and length 25 cm, with a deviation from linearity of less than 5% over this volume.