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We have made a systematic investigation of the Zeeman splitting of nϭ1 heavy-hole excitons in a range of Al 0.36 Ga 0.64 As/GaAs and In x Ga 1Ϫx As/GaAs ͑xϭ0.08 and 0.11͒ quantum wells at 1.8 K and in magnetic fields of up to 6 T applied along the growth axis ͑001͒. Calculations of splitting as a function of field were made using an eight-band K-P model which reproduce all the main features of the experimental data, including the sign, and give good quantitative agreement. The observed splittings are linear in low field ͑Ͻ1 T͒, but become nonlinear as field is increased. This behavior is attributed to a spin-dependent field-induced admixture between the light-and heavy-hole valence bands. For the GaAs/AlGaAs system agreement between experiment and theory requires a value for the Luttinger parameter in bulk GaAs close to 1.2 which is the generally accepted value, and rules out a lower value ͑0.7͒ which was proposed recently. From the theoretical fits to the In x Ga 1Ϫx As/GaAs Zeeman data we find that there is significant ''bowing'' of (x) which can be reproduced accurately using a perturbation theory relating the Luttinger and ␥ parameters, where ␥ 1,2,3 are obtained from experimentally determined light-and heavy-hole effective masses. ͓S0163-1829͑97͒05624-5͔

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