Omnidirectional mobile robots are capable of arbitrary motion in an arbitrary direction without changing the direction of wheels because they can perform 3-DOF motions on a plane. This paper presents a novel mobile robot design with steerable omnidirectional wheels. This robot can operate in either omnidirectional or differential drive modes, depending on the drive conditions. In the omnidirectional mode, the robot has 3 DOF in motion and 1 DOF in steering, which can function as a continuously variable transmission (CVT). The CVT function can be used to enhance the efficiency of the robot operation by increasing the range of the velocity ratio of the robot velocity to wheel velocity. The structure and kinematics of this robot are presented in detail. In the proposed steering control algorithm, the steering angle is controlled such that the motors may operate in the region of high velocity and low torque, thus operating with maximum efficiency. Various tests demonstrate that the motion control of the proposed robot works satisfactorily and the proposed steering control algorithm for CVT can provide a higher efficiency than the algorithm using a fixed steering angle. In addition, it is shown that the differential drive mode can give better efficiency than the omnidirectional-drive mode.