Game theory has been very successful in shaping modern economic theory over the past fifty years. Yet, the solution concepts developed under the assumption of perfect rationality require a degree of cognitive sophistication on players part that need not be realistic. In this project, I wish to broaden the definitions of equilibrium concepts to take into account the cognitive limitations of players. Armed with these equilibrium concepts, I wish to revisit a number of classic economic applications of game theory and economics in the hope that the proposed approach enhances our economic understanding. I also wish to check whether the proposed concepts are confirmed experimentally. Specifically, the project will rely on three new solution concepts I have recently introduced: the limited foresight equilibrium (Jehiel, 1995) in which players are viewed as knowing only the evolution of moves over the next n periods, the analogy-based expectation equilibrium (Jehiel, 2005) in which players understand only the average behavioural strategy of their opponents over bundles of states, and the valuation equilibrium (Jehiel and Samet, 2007) in which players attach the same valuation to a bundle of moves (possibly corresponding to different decision nodes). In each case, I assume that players choose their strategy based on the simplest representation of their environment that is consistent with their partial understanding. And as in the standard rationality paradigm, I assume that the partial understanding of players as parameterized by their cognitive type is correct. The heart of the project is to show how these approaches can be used to shed new light on major subfields of economic theory such as mechanism design, the theory of reputation, the theory of incomplete contracts and the theory of speculative markets. I also wish to test experimentally the solution concepts so as to check their empirical validity.