Quantum Schur–Weyl theory refers to a three-level duality relation. At Level I, it investigates a certain double centralizer property, the quantum Schur– Weyl reciprocity, associated with some bimodules of quantum gln and the Hecke algebra (of type A)—the tensor spaces of the natural representation of quantum gln (see [43], [21], [27]). This is the quantum version of the well-known Schur–Weyl reciprocity which was beautifully used in H. Weyl's influential book [77]. The key ingredient of the reciprocity is a class of important finite dimensional endomorphism algebras, the quantum Schur algebras or q-Schur algebras, whose classical version was introduced by I. Schur over a hundred years ago (see [69], [70]). At Level II, it establishes a certain Morita equivalence between quantum Schur algebras and Hecke algebras. Thus, quantum Schur algebras are used to bridge representations of quantum gln and Hecke algebras. More precisely, they link polynomial representations of quantum gln with representations of Hecke algebras via the Morita equivalence. The third level of this duality relation is motivated by two simple questions associated with the structure of (associative) algebras. If an algebra is defined by generators and relations, the realization problem is to reconstruct the algebra as a vector space with hopefully explicit multiplication formulas on elements of a basis; while, if an algebra is defined in terms of a vector space such as an endomorphism algebra, it is natural to seek their generators and defining relations.