In this paper, we investigate the adaptive control problem for robot manipulators with both the uncertain kinematics and dynamics. We propose two adaptive control schemes to realize the objective of task-space trajectory tracking irrespective of the uncertain kinematics and dynamics. The proposed controllers have the desirable separation property, and we also show that the first adaptive controller with appropriate modifications can yield improved performance, without the expense of conservative gain choice. The performance of the proposed controllers is shown by numerical simulations.