The usual economic perspective on a limit order emphasizes its role in supplying liquidity. We investigate the trading of 300 Nasdaq-listed stocks on the Island ECN, an electronic communication network organized as a limit order book. We find that a substantial portion of the limit orders are cancelled within an extremely brief time. We term "fleeting orders" those limit order that are cancelled within two seconds of submission, and explore the role they play in trading strategies. Our principal finding is that fleeting limit orders are closer substitutes for market orders than for traditional limit orders. Our results suggest that the aim of a trader who submits a fleeting order is to demand immediacy. This contrasts with the traditional view of limit order traders as patient providers of liquidity. We hypothesize that a "new equilibrium" has arisen, driven by improved technology, the emergence of an active trading culture, and increased market fragmentation. The new environment transforms the market from one in which prices are posted (visible limit orders) into one where searches (for hidden liquidity) are needed in order to achieve better terms of trade.