This paper presents a framework for the motion planning and control of redundant manipulators with the added task of collision avoidance. The algorithms that were previously studied and tested by the authors for planar cases are here extended to full mobility redundant manipulators operating in a three-dimensional workspace. The control strategy consists of a combination of off-line path planning algorithms with on-line motion control. The path planning algorithm is used to generate trajectories able to avoid fixed obstacles, detected before the robot starts to move; it is based on the potential fields method combined with a smoothing interpolation that exploits Bézier curves. The on-line motion control is designed to compensate for the motion of the obstacles and to avoid collisions along the kinematic chain of the manipulator; it is realized by means of a velocity control law based on the null space method for redundancy control. A term of the control law takes into account the speed of the obstacles as well as their position. In order to test the algorithms, a set of simulations are presented: the robot KUKA LBR iiwa is controlled in different cases, where fixed or dynamic obstacles interfere with its motion. Simulations are also used to estimate the required computational effort in order to verify the transferability to a real system.