In apparatus for purifying waste water with activated sludge deposited on carrier particles, in which air is introduced into a treating tank for circulating the waste water and the carrier particles in the tank through an air lift, a separator is provided at an upper end of the tank for discharging purified water together with sludge peeled off from the particles thus leaving bared carrier particles in the tank for recirculation. A portion of the bottom of the tank is inclined to prevent precipitation of the activated carrier particles. The tank has a rectangular cross-sectional configuration to enable to arrange a number of tanks in a juxtaposed relation in series or parallel in a narrow floor space.