The quantum cohomology ring of a symplectic manifold is a deformation of the usual cohomology ring. Since its structure constants are given by Gromov-Witten invariants, the quantum ring, too, is an invariant of the symplectic topology of the manifold. In this talk I will report on recent progress (by Li/Ruan, Ionel/Parker, and others) on the question how this quantum ring changes under symplectic cuts (or gluing, which is the inverse operation). In the case of Kähler manifolds, these operations include for example the blow-up along complex submanifolds.