Participation in networks inevitably involves risk. However, the study of networks has, perhaps surprisingly, not had much to say about network risk in the sense that most economists would use the term ‘risk.’ No consensus has even emerged on what such a model would constitute. Network risk appears to be present in the world, whether in the financial sector, in transportation, or with regards to interpersonal connections, and yet we have few tools for modeling it. The primary contribution of this thesis is a formal notion of network risk, and a set of tools for measuring it.