This paper introduces the notion of kinematic redundancy in nonholonomic mechanical systems, identifies some of the interesting properties which result because of the presence of the redundancy, and initiates a study of the control and application of these systems. It is shown that kinematic redundancy in nonholonomic system can be exploited both to simplify the problem of controlling these systems and to enhance their performance capabilities. Moreover, it is demonstrated that these results can be obtained even in the presence of considerable uncertainty regarding the system model. The proposed ideas are illustrated through the study of three example systems: a space robot, a mobile manipulator, and a tractor-trailer system with two steering inputs (fire truck).