A theory of phonon-assisted optical transitions in semiconductor quantum dots is developed which takes into account the non-adiabaticity of the exciton-phonon system. The role of non-adiabaticity is shown to be of paramount importance in spherical quantum dots, where the lowest one-exciton state can be degenerat4e or quasi-degenerate. In quantum dots of lower symmetry, the effects of non-adiabaticity reveal themselves mainly due to the phonon-induced mixing of states which belong to different energy levels. Our approach is applied to explain the optical spectra of several quantum-dot structures: ensembles of spherical CdSe, CdSe_xS_(1-x) and PbS quantum dots, self-assembled InAs/GaAs and CdSe/ZnSe quantum dots, brick-shaped InAs/GaAs quantum dots and CdS/HgS/CdS quantum dot heterostructures.
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