In the Localization game played on graphs, a set of cops uses distance probes to identify the location of an invisible robber. We present an extension of the game and its main parameter, the localization number, to oriented graphs. We present several bounds on the localization number of an oriented graph, including a tight bound via strong components, a bound using a linear programming problem on hypergraphs, and bounds in terms of DAG-width. A family of oriented graphs of order n is given with localization number (1−o(1))n/2. We investigate the localization number of random and quasi-random tournaments, including Paley tournaments.