Revenue Management (RM) is the practice of managing perishable assets by control-ling their availability and/or prices with an objective to maximize the total revenue. Seat inventory allocation falls in the purview of quantity-based RM. The liberalization of the aviation sector and the subsequent entrance of the low-cost carriers saw an ever-increasing customer base for the airline industry. Given the large number of buyers, firms were free to decide the price at which they would sell tickets. The low-cost carriers started to follow a third degree price discrimination and segmentation of the market, charging a higher price to the market with a relatively inelastic demand. Although a lot of work has been done in the area of seat inventory allocation under a monopolistic market scenario, we realized that not a lot of work had been done in a competitive market scenario. This thesis considers the problem of seat inventory allocation and pricing in a duopoly where each of the competing airlines have two fare-classes. We consider the possibility that the same fare-class may be priced differently by the two competing airlines and allow for the over flow of passengers between the airlines in the same fare-class. In the first part of our work, we develop a non-linear mathematical model for setting the booking limits for one of the two competing air-lines such that the revenue earned is maximized. We consider over flow of passengers from one airline to another in the same fare-class in response to a price differential and compare the results obtained from our model with the standard Expected Marginal Seat Revenue (EMSR) model under a monopolistic scenario. The results show that our model gives higher revenues than that obtained from the EMSR model. In the second part of our work, we consider a non-cooperative game between two competing airlines with price cutting as the strategy to increase their demand. Through numerical computations, we identify the pure strategy Nash equilibrium. From the results, we conclude that Nash equilibrium is achieved only when both the airlines follow the same pricing strategy indicating that individual price cutting will not be beneficial. This also indicates that unless the competitors enter into a cooperative coalition with each other, they would not benefit from deep discount offers. In the third and final part, we prove theoretically the existence of pure strategy Nash equilibrium in a two airline, two fare-class problem with price sensitive over flow of customers in the same fare-class that was computationally analysed earlier. The strategy / strategies at which Nash equilibrium is achieved are identified. We show that Nash equilibrium is only achieved when both the airlines price identically. Hence, our thesis concludes that differential pricing does not hold any significance for the competing airlines from an operational perspective.