Many problems have a structure with an inherently two (or higher) dimensional nature. Unfortunately, the classical method of representing problems when using Genetic Algorithms (GAs) is of a linear nature. We develop a genome representation with a related crossover mechanism which preserves spatial relationships for 2D problems. We then explore how crossover disruption rates relate to the spatial structure of the problem space. After discussing why a more appropriate representation is needed and exploring the theoretical aspects of our method, we empirically test our method to verify that it will be effective. We develop an easily understood abstracted class of problems with a 2D structure. A Monte Carlo study comparing the GAs using the string and matrix methods on a number of members of this problem class is then conducted. Results are presented which clearly show that for this particular problem, a matrix oriented GA should be used. Given our success in applying the matrix representation to an abstracted problem, we apply our methods to a real world image processing problem. We develop a method for using a GA with a matrix representation to denoise a greyscale image, and we apply this method to a noisy image. Finally, we discuss further ways in which to extend this work. Possible future image processing applications include various problems such as filter design, segmentation and edge detection. Other applications include semi-parametric density estimation, nonlinear multiple regression and solutions of multi-parameter multi-equation systems. We also discuss how problems where higher dimensional structures might be employed to further generalize our work to cases other than 2D problems.