This is an open access title available under the terms of a [CC BY-NC-ND 4.0 International] licence. It is free to read on the Oxford Academic platform and offered as a free PDF download from OUP and selected open access locations. Herbert Simon held that the fundamental turn in the study of bounded rationality is the turn from substantive to procedural rationality. Theories of substantive rationality begin with normative questions about attitudes: what should we prefer, intend, or believe? By contrast, theories of procedural rationality begin with normative questions about processes of inquiry: how should we determine what to prefer, intend, or believe? If Simon was right, then the central task for theories of bounded rationality is to develop an account of rational inquiry for bounded agents. We need, that is, a theory of inquiry under bounds.Inquiry Under Bounds takes as its starting point a five-point bounded rationality program inspired by recent work in cognitive science. To elaborate on and defend that program, Thorstad argues we need an account of rational inquiry for bounded agents. Inquiry under bounds develops an account of rational inquiry for bounded agents: the reason-responsiveness consequentialist view. I use this account to clarify and defend key insights from the bounded tradition as well as to shed light on recent controversies in the epistemology of inquiry.