We examine the economic analysis of the relationship between innovation and product market competition. First, we give a brief tour of the intellectual history of the area. Second, we examine how the Aghion-Howitt framework has influenced the development of the literature theoretically and (especially) empirically, with an emphasis on the “inverted U”: the idea that innovation rises and then eventually falls as the intensity of competition increases. Thirdly, we look at recent applications and development of the framework in the areas of competition policy, international trade and structural Industrial Organization.