Starting from classical problems in linear algebra and experiencing a spectacular development in the last forty years, the theory of representations of finite dimensional algebras is today one of the most popular and active fields in algebra. New theoretical approaches (like the use of quiver-theoretical techniques, the introduction of almost split sequences and Auslander-Reiten theory, the development of tilting theory, the notion of Gabriel-Roiter measure, the theory of Ringel-Hall algebras, the results in cluster theory) and the development of new computational algebra packages and tools (like GAP and Magma) have contributed not only to the deep understanding of the structure of module categories over some finite dimensional algebras, but have also connected representation theory with many other fields even outside algebra (for example quantum groups, algebraic geometry, control theory of linear dynamical systems).

The present monograph records the progress made by the authors in the last fifteen years regarding tame Ringel-Hall polynomials and their various applications, using a large spectrum of combinatorial, computational and representation theoretical tools.