A systematic investigation of the relationship between different redshift estimation schemes for more than 91000 quasars in the Sloan Digital Sky Survey (SDSS) Data Release 6 is presented. The publicly available SDSS quasar redshifts are shown to possess systematic biases of {Delta}z/(1+z)>=0.002 (600km/s) over both small ({delta}z~=0.1) and large ({delta}z~=1) redshift intervals. Empirical relationships between redshifts based on (i) CaII H&K host galaxy absorption, (ii) quasar [OII] {lambda}{lambda}3728, (iii) [OIII] {lambda}{lambda} 4960, 5008 emission and (iv) cross-correlation (with a master-quasar template) that includes, at increasing quasar redshift, the prominent MgII {lambda}{lambda} 2799, CIII] {lambda}{lambda} 1908 and CIV {lambda}{lambda} 1549 emission lines are established as a function of quasar redshift and luminosity. New redshifts in the resulting catalogue possess systematic biases, a factor of ~=20 lower compared to the SDSS redshift values; systematic effects are reduced to the level of {Delta}z/(1+z) (30km/s) per unit redshift or <=2.5x10^-5^ per unit absolute magnitude. Redshift errors, including components due both to internal reproducibility and to the intrinsic quasar-to-quasar variation among the population, are available for all quasars in the catalogue. The improved redshifts and their associated errors have wide applicability in areas such as quasar absorption outflows, quasar clustering, quasar-galaxy clustering and proximity-effect determinations.