A relationally exchangeable structure is a random combinatorial structure whose law is invariant with respect to relabeling its relations, as opposed to its elements. Historically, exchangeable random set partitions have been the best known examples of relationally exchangeable structures, but the concept now arises more broadly when modeling interaction data in modern network analysis. Aside from exchangeable random partitions, instances of relational exchangeability include edge exchangeable random graphs and hypergraphs, path exchangeable processes, and a range of other network-like structures. We motivate the general theory of relational exchangeability, with special emphasis on the alternative perspective it provides and its benefits in certain applied probability problems. We then prove a de Finetti-type structure theorem for the general class of relationally exchangeable structures.