  We study the quasi-two-dimensional Bose gas in harmonic traps at temperatures
above the Kosterlitz-Thouless transition, where the gas is in the normal phase.
We show that mean-field theory takes into account the dominant interaction
effects for experimentally relevant trap geometries. Comparing with Quantum
Monte Carlo calculations, we quantify the onset of the fluctuation regime,
where correlations beyond mean-field become important. Although the density
profile depends on the microscopic parameters of the system, we show that the
correlation density (the difference between the exact and the mean-field
density) is accurately described by a universal expression, obtained from
classical-field calculations of the homogeneous strictly two-dimensional gas.
Deviations from universality, due to the finite value of the interaction or to
the trap geometry, are shown to be small for current experiments. We further
study coherence and pair correlations on a microscopic scale. Finite-size
effects in the off-diagonal density matrix allows us to characterize the
cross-over from Kosterlitz-Thouless to Bose-Einstein behavior for small
particle numbers. Bose-Einstein condensation occurs below a characteristic
number of particles which rapidly diverges with vanishing interactions.
