An unpredictably maneuvering speedy target travels in a plane, which also hosts a team of fully actuated robots whose velocities and accelerations are upper-bounded in magnitude. The robots should approach the target and then follow it at a prespecified distance. They also should achieve an even self-distribution around the target and a given angular velocity of rotation about the target. Every robot has access to the relative position of the target and other robots (in the latter case, within a finite “visibility” range) and to the angular speed of its own pure rotation; access to the coordinates of its own linear velocity in its own local frame is also employed in some cases. The robots are not equipped with communication facilities and cannot distinguish among one another; assignment of different roles to various robots is infeasible. Necessary conditions for the mission feasibility are first obtained. A distributed control law is then presented, and its global convergence and collision avoidance property are rigorously justified under slight enhancement of the just mentioned necessary conditions. The performance of the control law is illustrated by computer simulations.