We have theoretically studied exciton states and photoluminescence spectra of strained wurtzite Alx Ga1-x N/GaN quantum-well heterostructures. The electron and hole energy spectra are obtained by numerically solving the Schrödinger equation, both for a single-band Hamiltonian and for a nonsymmetrical six-band Hamiltonian. The deformation potential and spin-orbit interaction are taken into account. For increasing built-in field, generated by the piezoelectric polarization and by the spontaneous polarization, the energy of size quantization rises and the number of size-quantized electron and hole levels in a quantum well decreases. The exciton energy spectrum is obtained using electron and hole wave functions and two-dimensional Coulomb wave functions as a basis. We have calculated the exciton oscillator strengths and identified the exciton states active in optical absorption. For different values of the Al content x, a quantitative interpretation, in good agreement with experiment, is provided for (i) the redshift of the zero-phonon photoluminescence peaks for increasing the quantum-well width, (ii) the relative intensities of the zero-phonon and one-phonon photoluminescence peaks, found within the nonadiabatic approach, and (iii) the values of the photoluminescence decay time as a function of the quantum-well width.