Most theories of curiosity emphasize the acquisition of information. Such conceptualizations focus on the actions of the knower in seeking units of knowledge. Each unit is valued as an unknown and appropriated in becoming known. Yet, recent advances across a range of disciplines from philosophy to cognitive science suggest that it may be time to complement the acquisitional theory of curiosity with a connectional theory of curiosity. This alternative perspective focuses on the actions of the knower in seeking relations among informational units, laying down lines of intersection, and thereby building a scaffold or network of knowledge. Intuitively, curiosity becomes edgework. In this chapter, we dwell on the notion of edgework, wrestle with its relation to prior accounts, and exercise its unique features to craft alternative reasons for curiosity's value to humanity. To begin, we engage in a philosophical discussion of the evidence for connectional curiosity across the last two millennia in the Western intellectual tradition. We then move to a contemporary operationalization of connectional curiosity in the mathematical language of network science. To make our discussion more concrete, we walk through a case study of humans browsing Wikipedia. The groundwork laid, we turn to the practical question of how (if at all) the paradigm of curiosity as edgework manifests in the contemporary lives of humans today. Does such a conceptualization help us to better understand the relations between curiosity and mental health? Might the edgework paradigm explain the drive to build specific structures of knowledge? Would the account help us to encode, test, and validate existing theories of curiosity, or propose new ones? Could it clarify why and how our culture values curiosity, in its multiple manifestations, plethora of practices, and kindred kinds in many bodies? In considering interdisciplinary answers to these questions, we find that the notion of edgework offers a fresh, flexible, and explanatory account of curiosity. More broadly, it uncovers new opportunities to use the lens of science to examine, probe, and interrogate this important dimension of the human experience.