Summary: The Schur-Weyl duality theorem motivates the general definition of dual representations. Such representations arise in a very large class of groups, including all compact, nilpotent and semidirect product groups, as well as the complementary groups in the physics literature and the reductive dual pairs in the mathematics literature. Several applications of the notion of duality are given, including the determination of polynomials and operator invariants, and the discovery of unusual dual algebras.