A magneto-optical study of accepters in varying degrees of confinement in GaAs/Al-x Ga-l-x, As quantum wells (QW's) is presented. A model for the splitting of the acceptor bound exciton (BE) and the allowed BE transitions in the presence of a magnetic field is presented. Our experimental results for the acceptor BE emission agree with calculated g values for bound holes, e.g., g(3/2) = 0. 61 and g(1/2) = 0.35 for a 150-Angstrom-wide QW. Furthermore, the energy separation between the acceptor m(j) = +/-3/2 1S(T-6) ground state and the excited m(j) = +/- 3/2 2S(T-6) state as a function of applied magnetic field is derived from selective photoluminescence and resonant-Raman-scattering measurements. Finally, these experimental results are compared with theoretical predictions for the confined acceptor states.