Large acoustic noise generated by magnetic resonance imaging (MRI) scanners pose significant problems for a patient undergoing a scan, in addition to attending medical staff. A heightened sense of anxiety, difficulty communicating with medical staff, and mild irritation through to acute discomfort are all results of the large noise amplitude produced during the scanning process. Gradient coils are a component of MRI hardware that have been found to be the major contributor of acoustic noise and are thus the subject of this thesis. A gradient coil is required to produce a homogeneous linear field to excite nuclei in a predictable manner so that clear images can be obtained. A feature of gradient coils is that they must be switched on and off repeatedly during the imaging process. This switching generates large Lorentz forces on the chamber, which causes the chamber to deform and promotes a pressure wave inside the chamber that can be heard as acoustic noise. The problem becomes one of designing current winding patterns on the gradient coil that produce a specified linear field and have a reduced simulated acoustic noise output. This problem is studied in detail and is accomplished using a Tikhonov regularisation process. This thesis looks directly at theoretical methods to actively reduce simulated acoustic noise during the MRI scanning process. Pro-active noise reduction, in this work, is achieved by reducing the deflection of the gradient coil, which leads to the design of quiet gradient coils. Beginning with a relatively simple one dimensional radial coil deformation model and requiring a high level of linearity of the gradient field, designs for the current windings on the gradient coils are produced. In this first model a comparatively small reduction of 0.6 dB was attained; however, previous work was extended to account for realistic switching sequences. Next, a more sophisticated three dimensional deflection model was considered, which produced a similar small level of noise reduction, but nonetheless substantial insight into the physical process behind the generation of noise was obtained. Finally, a robust technique for designing very quiet gradient coils, reducing the noise by 49 dB, was achieved by abandoning the requirement that the gradient