Research reported in this article has been conducted from the theoretical perspective of the Anthropological Theory of the Didactic developed by Y. Chevallard and his collaborators, and from the institutional perspective of university mathematics education in France. It focuses on the teaching and learning of algebraic structures. The article introduces the notion of structuralist praxeology in the theory of teaching and learning of Abstract Algebra. The notion is illustrated by detailed descriptions of structuralist praxeologies related to the arithmetic of abstract rings, identified, on the one hand, in the history of mathematics and in Abstract Algebra textbooks and, on the other, in interactions among a group of learners of Abstract Algebra on an online discussion forum. The theoretical (and practical) issues regarding the teaching and learning of the structuralist praxeologies are discussed in the context of the difficulties that learners experience in the transition from undergraduate and graduate studies in mathematics at university. The article concludes with an outline of a research program based on further praxeological analyses of the tasks, methods, methodologies and theories characteristic of structuralist thinking in a teaching and learning environment.