The focus of this work is path planning algorithms for autonomous agents. Specifically, we study problems in three areas where path planning to direct the motion of autonomous agents is critical for their performance. The first problem is a vehicle routing problem in which mobile demands appear in an environment and the task of the autonomous agent is to stop the demands from escaping the environment boundary. We first propose two fundamental performance bounds for the proposed problem. We then propose routing algorithms for this problem with performance guarantees. We examine the gap between these guarantees and the fundamental performance bounds. The second problem is a surveillance problem in a networked environment. The tasks of the autonomous surveillance agent in this problem are to (1) detect unknown intruder locations and (2) detect anomalies based on noisy measurements. We propose Markov chain based routing algorithms for the surveillance agent to achieve these goals. We parameterize these routing algorithms using a property of Markov chains called the mean first passage time. We also frame optimization problems to obtain optimal algorithms for the two surveillance tasks. The third problem studied in this work is a boundary guarding problem in which the task of a set of patrolling agents constrained to move on a ring is to achieve synchronization using only local communication. We propose a coordination algorithm to solve this problem and identify initial agent configurations under which synchronization is guaranteed.