There is a large body of literature that concludes that - when confronted with increased competition - banks rationally choose more risky portfolios. We argue that this literature has had a significant influence on regulators and central bankers. We review the empirical literature and conclude that the evidence is best described as "mixed." We then show that existing theoretical analyses of this topic are fragile, since there exist fundamental risk-incentive mechanisms that operate in exactly the opposite direction, causing banks to become more risky as their markets become more concentrated. These mechanisms should be essential ingredients of models of bank competition.