In the present work we study the defect states in recently synthesized semiconductor quantum dot–quantum well systems (QDQWs). We employ the effective-mass theory (EMT) with a realistic barrier and variable effective mass. The model is simple and all of our results are obtained by an exact numerical diagonalization of the Schrödinger equation. We study the ground state of the host system as a function of quantum well size. We demonstrate that the upshift with QDQW downsizing differs from reported upshifts in simple quantum dots (QDs) and explain this by means of a perturbative analysis. We study the binding energy of the hydrogenic impurity and its variation with QDQW size. Next the binding energy is studied as the impurity is moved off-centre. We find that the binding energy goes through a maximum. An analysis of the wavefunction is carried out to obtain an understanding of this surprising effect. The impurity calculations are carried out on CdS/HgS/CdS QDQWs. Recently, experimental studies on a monolayer of HgS in a CdS dot were carried out. We model this system as a `δ-defect' consisting of a thin spherical shell of fixed potential depth in a spherical CdS dot. The ground state of this system is studied as the shell is dragged from the periphery to the centre of the CdS dot. Our results are explained on the basis of qualitative arguments and asymptotic analyses. Outstanding issues and future directions are suggested.