The source and nature, as well as the history of ray-effects, is described. A benchmark code, using piecewise constant functions in angle and diamond differencing in space, is derived in order to analyze four sample problems. The results of this analysis are presented showing the ray effects and how increasing the resolution (number of angles) eliminates them. The theory of wavelets is introduced and the use of wavelets in multiresolution analysis is discussed. This multiresolution analysis is applied to the transport equation, and equations that can be solved to calculate the coefficients in the wavelet expansion for the angular flux are derived. The use of thresholding to eliminate wavelet coefficients that are not required to adequately solve a problem is then discussed. An iterative sweeping algorithm, called the SN-WN method, is derived to solve the wavelet-based equations. The convergence of the SN-WN method is discussed. An algorithm for solving the equations is derived, by solving a matrix within each cell directly for the expansion coefficients. This algorithm is called the CWWN method. The results of applying the CW-WN method to the benchmark problems are presented. These results show that more research is needed to improve the convergence of the SN-WN method, and that the CW-WN method is computationally too costly to be seriously considered.