The research presented in this dissertation pertains to the development of a real time SLAM solution that can be performed by a low cost autonomous underwater vehicle equipped with low cost and memory constrained computing resources. The design of a custom rangefinder for underwater applications is presented. The rangefinder makes use of two laser line generators and a camera to measure the unknown distance to objects in an underwater environment. A visual odometry algorithm is introduced that makes use of a downward facing camera to provide our underwater vehicle with localization information. The sensor suite composed of the laser rangefinder, downward facing camera, and a digital compass are verified, using the Extended Kalman Filter based solution to the SLAM problem along with the particle filter based solution known as FastSLAM, to ensure that they provide in- formation that is accurate enough to solve the SLAM problem for out low cost underwater vehicle. Next, an extension of the FastSLAM algorithm is presented that stores the map of the environment using an occupancy grid is introduced. The use of occupancy grids greatly increases the amount of memory required to perform the algorithm so a version of the Fast- SLAM algorithm that stores the occupancy grids using the Haar wavelet representation is presented. Finally, a form of the FastSLAM algorithm is presented that stores the occupancy grid in compressed form to reduce the amount memory required to perform the algorithm. It is shown in experimental results that the same result can be achieved, as that produced by the algorithm that stores the complete occupancy grid, using only 40% of the memory required to store the complete occupancy grid.