  We discuss the quasi two-dimensional trapped Bose gas where the thermal
occupation of excited states in the tightly confined direction is small but
remains finite in the thermodynamic limit. We show that the semiclassical
theory describes very accurately the density profile obtained by Quantum Monte
Carlo calculations in the normal phase above the Kosterlitz--Thouless
temperature T_KT, but differs strongly from the predictions of strictly
two-dimensional mean-field theory, even at relatively high temperature. We
discuss the relevance of our findings for analyzing ultra-cold-atom experiments
in quasi two-dimensional traps.
