This dissertation uses the notion of collision cones and rendezvous cones to address several motion planning problems for autonomous vehicles. Collision avoidance is fundamental to the problem of planning safe trajectories in dynamic environments. This problem appears in several diverse elds including robotics, air vehicles, underwater vehicles and computer animation. In the rendezvous problem, generating appropriate trajectories to achieve overlap of footprints of unmanned aerial vehicles is important in problems related to search and surveillance, and for establishing communication between a network of UAVs, or between a user and a base station in remote areas. In the collision avoidance problem, much of the collision avoidance literature assumes shapes of the objects as circles. However, when objects are operating in closer proximity, or when objects are elongated and/or have non-convex shapes, a less conservative approach, that considers the exact shapes of the objects, is more desirable. This dissertation presents analytical collision avoidance laws in cooperative and non-cooperative dynamic environments. The collision avoidance laws are simulated on Ionic Polymer-Metal Composite (IPMC) actuated robotic sh. Collision cones are also used to analyze pursuit evasion games between two objects of arbitrary shapes. Collision avoidance of objects that can deform by changing their shape as a function of time is also presented. The rendezvous problem requires communication/sensing footprints of vehicles to overlap. The need of the footprints to overlap is dictated by the requirement that no part of the sensed area is left uncovered in a search and surveillance operation; or by the need to position a relay UAV in the overlap region of two distant UAVs in order to enable them to communicate with each other. The concept of a rendezvous cone is used as the basis for the development of nonlinear analytical guidance laws that enable the overlap of footprints to the requisite depth.