Throughout proof theory, proofs are often taken to be well-founded (often finite) trees of inferences. Theories of inductive definitions, among other theories, bump up against the limitations of this perspective, and a variety of formalisms have been used to push beyond this – Girard’s beta-proofs, non-well-founded proofs, proofs-as-functions. We briefly describe some features of these approaches, the way these perspectives are essentially equivalent, and the way well-foundedness reasserts itself as a core property of proofs.