A well-studied problem in the literature on airline revenue (or yield) management is the optimal allocation of seat inventory among fare classes, given a demand distribution for each class. In practice, the seat allocation decisions of one airline affect the passenger demands for seats on other airlines. In this paper, we examine the seat inventory control problem under both horizontal competition (two airlines compete for passengers on the same flight leg) and vertical competition (different airlines fly different legs on a multileg itinerary). Such vertical competition can be the outcome of a code-sharing agreement between airlines, because each airline sells seats on the partner airlines’ flights but the airlines are unwilling, or unable, to coordinate yield management decisions. We provide a general sufficient condition under which a pure-strategy Nash equilibrium exists in these revenue management games, and we also compare the total number of seats available in each fare class with, and without, competition. Analytical results as well as numerical examples demonstrate that more seats are protected for higher-fare passengers under horizontal competition than when a single airline acts as a monopoly. Under vertical competition the booking limit may be higher or lower, however, than the monopoly level, depending on the demand for connecting flights in each fare class. Finally, we discuss revenue-sharing contracts that coordinate the actions of both airlines.