The history of Penrose diagrams in the physics of General Relativity (GR) is presented. It is argued that the diagrams did conceptual work for physicists, providing a literal place for abstract, formal objects. Penrose diagrams were associated with the mathematics of conformal transformations applied to GR. Together the diagrams and formalism reconfigured the basic concepts of the field—notions of space, time, cosmology, and energy. Nor were the meanings of the diagrams themselves stable over time. Their physical and conceptual evolution is traced. This history also demonstrates the tight integration of the contexts of research and pedagogy in the period investigated (1962–66). Diagrams circulated rapidly between research talks and publications and the pedagogical context of summer school lectures for advanced graduate students. Further reception and circulation of the diagrams is briefly examined.