We study the stability and efficiency of social and economic networks, when self-interested individuals have the discretion to form or sever links. First, in the context of two stylized models, we characterize the sets of stable networkds (immune to incentives to form or sever links) and the sets of efficient networks (those which maximize total production or utility). The sets of stable networks and efficients networks do not always intersect. Next, we show that this tension is not unique to these models, but persists generally. In order to assure that there is always at least one efficient graph which is stable, one is forced to allocate resources to nodes (players) who are not responsible for any of the production. We characterize another rule which fails to assure that efficient graphs are stable, but arises naturally if the allocations result from the bargaining of players.