A multistage pump bleaching process comprising at least three bleaching stages including oxygen extraction (20) in an alkaline medium (EO) of a chlorine (C) or chlorine/chlorine-dioxide (CD) bleaching step (5) followed by a hot alkaline hypochlorite treatment (H*/E), (28) or followed by a chlorine dioxide (D) treatment with simultaneous treatment with chlorine dioxide (H*/ED) in the sequence EOH*/ED or EO(H*/ED). The novel bleaching sequences result in significant savings in time and operating costs as compared with conventional bleaching sequences capable of producing products of comparable brightness.