This article introduces the concept of duality in financial networks. In bankruptcy problems, in which a bankrupt entity divides its non-negative assets among a group of claimants, duality of bankruptcy rules entails the division of losses versus gains. Financial networks generalize bankruptcy problems by allowing for multiple agents with individual assets interconnected by mutual claims. We show that allowing for negative assets is imperative to adequately formulate dual financial networks and dual bankruptcy problems. We show that there is a one-to-one correspondence between payment schemes based on bankruptcy rules in a financial network and payment schemes based on the dual of those bankruptcy rules in the dual financial network. Moreover, dual financial networks enable us to define dual transfer rules and dual allocation rules. We show that transfer rules based on self-dual bankruptcy rules need not necessarily be self-dual, whereas allocation rules based on self-dual bankruptcy rules are always self-dual.