Tactile sensing is an essential capability for a robot to perform manipulation tasks in cluttered environments. While larger areas can be assessed instantly with cameras, Lidars, and other remote sensors, tactile sensors can reduce their measurement uncertainties and gain information of the physical interactions between the objects and the robot end-effector that is not accessible via remote sensors. In this paper, we introduce the novel tactile sensor GelTip that has the shape of a finger and can sense contacts on any location of its surface. This contrasts to other camera-based tactile sensors that either only have a flat sensing surface, or a compliant tip of a limited sensing area, and our proposed GelTip sensor is able to detect contacts from all the directions, like a human finger. The sensor uses a camera located at its base to track the deformations of the opaque elastomer that covers its hollow, rigid, and transparent body. Because of this design, a gripper equipped with GelTip sensors is capable of simultaneously monitoring contacts happening inside and outside its grasp closure. Our extensive experiments show that the GelTip sensor can effectively localize these contacts at different locations of the finger body, with a small localization error of approximately 5 mm on average, and under 1 mm in the best cases. Furthermore, our experiments in a Blocks World environment demonstrate the advantages, and possibly a necessity, of leveraging all-around touch sensing in manipulation tasks. In particular, the experiments show that the contacts at different moments of the reach-to-grasp movements can be sensed using our novel GelTip sensor.