Applications for autonomous robots have long been identified in challenging environments including built-up areas, mines, disaster scenes, underwater and in the air. Robust solutions to autonomous navigation remain a key enabling issue behind any realistic success in these areas. Arguably, the most successful robot navigation algorithms to-date, have been derived from a probabilistic perspective, which takes into account vehicle motion and terrain uncertainty as well as sensor noise. Over the past decades, a great deal of interest in the estimation of an autonomous robot’s location state, and that of its surroundings, known as Simultaneous Localization And Map building (SLAM), has been evident. This presentation will explain recent advances in the representations of robotic measurements and the map itself, and their consequences on the robustness of SLAM. Fundamentally, the concept of a set based measurement and map state representation allows all of the measurement information, spatial and detection, to be incorporated into joint Bayesian SLAM frameworks. Representing measurements and the map state as random sets, rather than the traditionally adopted random vectors, is not merely a triviality of notation. It will be demonstrated that a set based framework circumvents the necessity for any fragile data association and map management heuristics, which are necessary, and often the cause of failure, in vector based solutions. This presentation will mathematically demonstrate that the set and vector based formulations are actually closely related, and that RFS SLAM can be viewed as a generalization of vector-based SLAM. Under ideal detection conditions, the two methods are equivalent. The findings provide important insights into some of the limitations of the random vector formulation. These are validated using Probability Hypothesis Density (PHD) Filter simulations with varying detection statistics, along with actual SLAM datasets and experiments demonstrating SLAM with laser and radar sensors in urban and marine environments. Comparisons of PHD Filter based SLAM and state of the art vector based implementations will demonstrate the robustness of the former to the realistic situations of sensor false alarms, missed detections and clutter.