ABSTRACTWe classify the shift operators for the symmetric Askey–Wilson polynomials and construct shift operators for the nonsymmetric Askey–Wilson polynomials using two decompositions of nonsymmetric Askey–Wilson polynomials in terms of symmetric ones. These shift operators are difference–reflection operators, and we discuss the conditions under which they restrict to shift operators for the symmetric Askey–Wilson polynomials. We use them to compute the norms of the nonsymmetric Askey–Wilson polynomials and compute their specializations for . These turn out to be shift operators for the nonsymmetric Heckman–Opdam polynomials of type that have recently been found.