We argue that investigating the biological mechanisms underlying the sensation of pain in humans and animals may lead to fundamental new insights about robot cognition, motor skill acquisition, autonomy, memory, and system integration. Despite the fact that pain plays a central role in the life of humans and more complex animals, it has received only peripheral attention in the field of robotics. In this paper, we discuss the complex web of mechanisms and functions underlying biological pain sensation and anticipation. Next, we examine the opportunities and challenges that arise when studying computational frameworks that mimic nociceptive pathways. Further, we propose two initial benchmark tasks that may be leveraged to accelerate such research. Our main objectives are to highlight a critical knowledge gap in our understanding of intelligent physical systems and to identify a new and promising avenue for further research.