This paper presents two hybrid strategies for robot visual servoing. Two specific image constraints, the image singularities and image local minima, are considered in both strategies. The hybrid motion control strategy consists of a local switching control between the image-based and position-based visual servoing for direct avoidance of image singularities and image local minima. The hybrid motion planning strategy consists of an artificial potential field-based global hybrid trajectory planner, where a complete set of Cartesian, image, and robot joint constraints under a complex visual servoing scenario are considered. In this strategy, the image singularities are resolved using the damped-least-square-based joint trajectory planning, while the image local minima are evaluated only along the planned image trajectories and automatically avoided in the image-based trajectory tracking. Two global planning methods are considered. In the first method, the end-effector trajectory is directly planned with respect to the stationary target object frame, which provides a much shorter translational path compared with the local planning method. In the second method, the target trajectory is planned with respect to the current end-effector frame, which minimizes the chances of image trajectories leaving the camera field of view. Simulation and experimental results are given to demonstrate the efficiency of the two hybrid strategies.