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This paper focuses on the problem of patrolling an environment with a team of autonomous agents. Given a set of strategically important locations (viewpoints) with different priorities, our patrolling strategy consists of 1) constructing a tour through the viewpoints, and 2) driving the robots along the tour in a coordinated way. As performance criteria, we consider the weighted refresh time, i.e., the longest time interval between any two visits of a viewpoint, weighted by the viewpoint's priority. We consider the design of both optimal trajectories and distributed control laws for the robots to converge to optimal trajectories. First, we propose a patrolling strategy and we characterize its performance as a function of the environment and the viewpoints priorities. Second, we restrict our attention to the problem of patrolling a nonintersecting tour, and we describe a team trajectory with minimum weighted refresh time. Third, for the tour patrolling problem and for two distinct communication scenarios, namely the Passing and the Neighbor-Broadcast communication models, we develop distributed algorithms to steer the robots toward a minimum weighted refresh time team trajectory. Finally, we show the effectiveness and robustness of our control algorithms via simulations and experiments.