  The action of the mapping class group of a surface on the collection of
homotopy classes of disjointly embedded curves or arcs in the surface is
discussed here as a tool for understanding Riemann's moduli space and its
topological and geometric invariants. Furthermore, appropriate completions,
elaborations, or quotients of the set of all such homotopy classes of curves or
arcs give for instance Thurston's boundary for Teichm\"uller space or a
combinatorial description of moduli space in terms of fatgraphs. Related open
problems and questions are discussed.
