The simulation of global illumination is one of the most fundamental problems in computer graphics, with applications is a wide variety of areas. This problem studies the light energy transfer between reflective surfaces in an environment. Initially derived from the field of thermal engineering, radiosity has emerged over the past several years as one of the most promising solution methods. Despite having produced some of the most realistic-looking computer generated images to date, radiosity methods have not yet met with widespread acceptance. The main obstacle has been their need for very careful and time consuming user intervention, without which, current techniques are prone to generating a wide range of annoying visual artifacts. These artifacts are generally due to poor surface meshing, resulting in insufficient sampling density and ineffective sample placement. This thesis investigates the roots of this problem by taking a step back from the traditional finite element formulation of radiosity and examining the more general integral equation formulation. An analysis of the radiance functions described by this equation shows how umbra and penumbra boundaries as well as other sharp changes in illumination actually correspond to discontinuities in the radiance function and its derivatives. The results of this analysis have led to the concept of discontinuity meshing, whereby accurate approximations to the radiance functions are computed by explicitly representing their discontinuities as boundaries in the mesh. This concept has been applied to the design of a discontinuity meshing algorithm for polyhedral environments. The algorithm is embedded in a progressive refinement radiosity system and uses piecewise quadratic interpolation to reconstruct a smooth radiance function while preserving discontinuities where appropriate. The radiosity solutions produced by the new algorithm are compared against a photograph of a physical environment, an analytical solution, and a conventional, yet state-of-the-art, radiosity system, and its performance on architectural models of medium complexity is measured. The results are remarkably accurate both numerically and visually. The new discontinuity meshing algorithm drastically reduces, and in many cases eliminates, many of the annoying artifacts typical of conventional radiosity meshes, producing images of previously unattained quality. Moreover, the meshing is completely automatic and produces solutions that are highly view-independent.