A theory of photoluminescence in semiconductor quantum dots is developed which relies on two key ingredients. First, it takes into account non-adiabaticity of the exciton-phonon system. Second, it includes the multimode dielectric model of LO-phonons and of the electron-phonon interaction in confined systems. The role of non-adiabaticity is shown to be of paramount importance in spherical quantum dots, where the lowest one-exciton state can be degenerate or quasi-degenerate. For various quantum dot structures, the calculated intensities of phonon satellites as a function of temperature, excitation energy and size of quantum dots are in a fair agreement with experimental data on photoluminescence.