We have shown recently that the magnetic properties of excitons change significantly as the excitons acquire kinetic energy. In particular, the exciton magnetic moments are enhanced considerably, whilst the diamagnetism decreases. The behaviour can be investigated through spectroscopic studies of excitons confined in quantum wells of large width (greater than five times the exciton Bohr radius) and these motion-induced changes in the magnetic properties have now been observed for CdTe, ZnSe, ZnTe and GaAs. The present paper summarises these phenomena, with particular focus on CdTe and ZnSe, and shows that the changes can be accounted for by motion-induced mixing between the exciton ground and higher lying states. The mixing is caused by the g 3 term in the Luttinger Hamiltonian which describes the dispersion curves for the valence band and, as a result, the form of the exciton wavefunction becomes motion-dependent. For both materials, excellent agreement is obtained between experiment and the results predicted by this mechanism.ß 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction What happens to excitons when they move through semiconductors? To answer this question is not easy, since the usual method of investigating the structure of excitons is to use optical spectroscopy and, unfortunately, the small wavevector of photons means that only excitons with low values of their translational wavevectors can conventionally be studied. To overcome this difficulty, we have in a series of recent investigations [1-6] studied excitons in potential wells which are sufficiently wide for quantum confinement energies to be smaller than the exciton binding energy. In such quantum wells, the exciton can be described in the adiabatic or centre of mass (CoM) approximation [7,8], in which it is considered as a composite particle formed from the electron and the hole mutually orbiting each other. The translational wavevector K z of the exciton as a whole is then quantised approximately according to K z ¼ np=L, where L is the width of the quantum well and N is a non-zero integer, and the recombination transition energies of the exciton then have discrete values. As a result of the loss in translational symmetry, photons can interact with excitons with large