Four emission bands, all belonging to the main thallium centres, have been detected in CsI:Tl, and their spectral, polarization and kinetic characteristics have been studied at 0.35-320 K. A new model is proposed for the excited states responsible for the luminescence of thallium-doped caesium iodide. In this model, the main thallium centre is considered as a cluster consisting of a Tl+ ion and at least 12 I- and 11 Cs+ ions. Two weak ultraviolet emission bands of CsI:TI (3,31 and 3.09 eV) are ascribed to electronic transitions from trigonal and tetragonal Jahn-Teller minima of the tripler relaxed excited state, whose structure is similar to that of Tl+ centres in Fee alkali halides. Two intense visible bands (2.55 and 2.25 eV) are assumed to arise from two different ('weak' and 'strong') off-centre configurations of a self-trapped exciton perturbed by the Tl+ ion. The minima responsible for all the emission bands are located on the same adiabatic potential energy surface. The excitonic-like nature of visible emission could explain the high scintillation efficiency of CsI:Tl.