Abstract: For the biological purification of waste water that contains organic and nitrogen-containing compounds, wastewater is aerated with air and/or oxygen in the presence of a biomass fixed on a carrier material 9 in a reactor 1 through which flow takes place lengthwise and then is separated in a secondary clarification 5 into purified water and sludge. The separated sludge is at least partially returned again to the reactor 1. Carrier particles in an amount that can move freely in the waste water are introduced into the reactor as carrier material 9 for the microorganisms. To enable the decomposition of both the organic compounds and the nitrogen-containing compounds simultaneously. In the reactor 1, a cylindrical flow 10 of the mobile reactor contents is produced by aeration perpendicular to the flow direction in the reactor 1.