Magnetic Resonance Imaging (MRI) is a popular medical diagnostic tool for imaging soft-tissue structures within the human body. A very particular combination of magnetic fields is required for the imaging process and these are generated by various coil systems in the scanner. Gradient coils are used to spatially-encode the image region and are typically constructed from copper wires wrapped around the surface of the cylinder within which the patient lies. High image resolution demands strong coil current and this can result in considerable Ohmic heating in regions of the coil where the windings are closely spaced, Of particular concern are gradient hot spots, which can cause system failure or image distortion. An analytical model is presented for predicting the spatial steady-state temperature distribution for cylindrical gradient coils under a range of geometry and material property considerations. Simulations are shown to compare well to experimentally measured temperature distributions for two coils of inherently different winding structure. In addition, a non-linear optimisation technique is presented for redesigning the gradient coils to display lower hot spot temperature at no cost to coil performance.