Robot-to-human object handover is an important step in many human robot collaboration tasks. A successful handover requires the robot to maintain a stable grasp on the object while making sure the human receives the object in a natural and easy-to-use manner. We propose ContactHandover, a robot to human handover system that consists of two phases: a contact-guided grasping phase and an object delivery phase. During the grasping phase, ContactHandover predicts both 6-DoF robot grasp poses and a 3D affordance map of human contact points on the object. The robot grasp poses are re-ranked by penalizing those that block human contact points, and the robot executes the highest ranking grasp. During the delivery phase, the robot end effector pose is computed by maximizing human contact points close to the human while minimizing the human arm joint torques and displacements. We evaluate our system on 27 diverse household objects and show that our system achieves better visibility and reachability of human contacts to the receiver compared to several baselines. More results can be found on https://clairezixiwang.github.io/ContactHandover.github.io