  We investigate Bose-Einstein condensation for ultracold bosonic atoms in
two-dimensional systems. The functional renormalization group for the average
action allows us to follow the effective interactions from molecular scales
(microphysics) to the characteristic extension of the probe $l$ (macrophysics).
In two dimensions the scale dependence of the dimensionless interaction
strength $\lambda$ is logarithmic. Furthermore, for large $l$ the frequency
dependence of the inverse propagator becomes quadratic. We find an upper bound
for $\lambda$, and for large $\lambda$ substantial deviations from the
Bogoliubov results for the condensate depletion, the dispersion relation and
the sound velocity. The melting of the condensate above the critical
temperature $T_c$ is associated to a phase transition of the
Kosterlitz-Thouless type. The critical temperature in units of the density,
$T_c/n$, vanishes for $l\to\infty$ logarithmically.
