This paper proposes a normative taxonomy by which universities can express the extent to which they meet five core epistemic responsibilities. Epistemic responsibilities are responsibilities that have to do with the attainment of knowledge and understanding. The core epistemic responsibilities, which we call the Big Five, are to (1) foster research integrity, (2) teach for intellectual virtue, (3) address the big questions of life, (4) give humanistic inquiry and education a proper place, and (5) serve society. The paper characterizes the Big Five in some detail and explains why they are core epistemic responsibilities of universities. The paper concludes by describing the steps that should be taken in order to test, amend, and implement the taxonomy.