Abstract: To achieve a successful grasp, gripper attributes such as its geometry and kinematics play a role as important as the object geometry. The majority of previous work has focused on developing grasp methods that generalize over novel object geometry but are specific to a certain robot hand. We propose UniGrasp, an efficient data-driven grasp synthesis method that considers both the object geometry and gripper attributes as inputs. UniGrasp is based on a novel deep neural network architecture that selects sets of contact points from the input point cloud of the object. The proposed model is trained on a large dataset to produce contact points that are in force closure and reachable by the robot hand. By using contact points as output, we can transfer between a diverse set of multifingered robotic hands. Our model produces over 90% valid contact points in Top10 predictions in simulation and more than 90% successful grasps in real world experiments for various known two-fingered and three-fingered grippers. Our model also achieves 93%, 83% and 90% successful grasps in real world experiments for an unseen two-fingered gripper and two unseen multi-fingered anthropomorphic robotic hands.