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Summary form only given. For over 15 years, a major research theme in robotics has been the development of randomized sampling schemes to create efficient motion planners. The main outcome of this research has been the Probabilistic Roadmap approach (PRM) to motion planning. Originally, this approach was intended to compute collision-free paths of robots with ldquomanyrdquo degrees of freedom-at that time, 4 or more. But, over the years, successive improvements (as well as faster computers) made it possible to handle robotic systems with several dozen degrees of freedom operating in complex geometric environments. PRM was also extended to solve planning problems with motion constraints other than collision avoidance, for instance, visibility, equilibrium, contact, and kinodynamic constraints. Concurrently, PRM planning has also been applied to non-robotics applications, e.g., for animating autonomous digital characters, designing product that can easily be assembled and serviced, testing whether architectural designs satisfy building codes, providing interactive tools to navigate in huge virtual reality models, planning complex surgical operations, and studying folding and binding molecular motions. This talk will consist of two parts. First, the PRM approach and various underlying techniques, especially sampling strategies, will be reviewed. Rather than computing an exact representation of a robotpsilas configuration space, the PRM approach ldquolearnsrdquo incrementally a simplified representation of the connectivity of this space. This representation is a network of simple paths connecting configurations sampled at random. The second part of the talk will analyze the necessary and sufficient conditions under which the PRM approach works well. It will argue that the main outcome of the research on PRM planning is what the empirical success of this approach tells us about motion planning problems, rather than the approach itself.