The electron effective g factor tensor in asymmetric III-V semiconductor quantum wells (AQWs) and its tuning with the structure parameters and composition are investigated with envelope-function theory and the ´k p 8 8 • Kane model. The spin-dependent terms in the electron effective Hamiltonian in the presence of an external magnetic field are treated as a perturbation and the g factors * ĝ and *  g , for the magnetic field in the QW plane and along the growth direction, are obtained analytically as a function of the well width L. The effects of the structure inversion asymmetry (SIA) on the electron g factor are analyzed. For the g-factor main anisotropy * * D = -^ AQWs are presented and discussed with the available experimental data; in particular InAs QWs are shown to not only present much larger g factors but also a larger g-factor anisotropy, and with the opposite sign with respect to GaAs QWs.