Luminosity functions and their integrated luminosity densities are presented for the 6dF Galaxy Survey (6dFGS). This ongoing survey ultimately aims to measure around 150,000 redshifts and 15,000 peculiar velocities over almost the entire southern sky at |b|>10 deg. The main target samples are taken from the 2MASS Extended Source Catalog and the SuperCOSMOS Sky Survey catalogue, and comprise 138,226 galaxies complete to (K, H, J, rF, bJ) = (12.75, 13.00, 13.75, 15.60, 16.75). These samples are comparable in size to the optically-selected Sloan Digital Sky Survey and 2dF Galaxy Redshift Survey samples, and improve on recent near-infrared-selected redshift surveys by more than an order of magnitude in both number and sky coverage. The partial samples used in this paper contain a little over half of the total sample in each band and are ~90 percent complete. Luminosity distributions are derived using the 1/Vmax, STY and SWML estimators, and probe 1 to 2 absolute magnitudes fainter in the near-infrared than previous surveys. The effects of magnitude errors, redshift incompleteness and peculiar velocities have been taken into account and corrected throughout. Generally, the 6dFGS luminosity functions are in excellent agreement with those of similarly-sized surveys. Our data are of sufficient quality to demonstrate that a Schechter function is not an ideal fit to the true luminosity distribution, due to its inability to simultaneously match the faint end slope and rapid bright end decline. Integrated luminosity densities from the 6dFGS are consistent with an old stellar population and moderately declining star formation rate.