A method for bleaching cellulosic and lignocellulosic pulp using hydrogen peroxide as a bleaching reagent, utilizes a two-step reaction, viz. a short high temperature step at a pressure sufficient to suppress boiling and a longer atmospheric pressure reduced temperature step to consume a substantial portion of the hydrogen peroxide remaining after the short phase. The method includes the steps of introducing pulp (15), at a consistency of 10 %-18 %, to a mixer (100) in which the pulp is heated to a temperature above 100 DEG C; adding sufficient sodium hydroxide to bring the pulp to a pH of 8.5; adding sufficient hydrogen peroxide (36) to equal from about 0.5 %-5.0 %, by weight, of the pulp; passing the pulp through a pressurized reactor column (400) at a rate providing a reaction time in the column of between 1 and 30 minutes; and depositing the pulp in a reaction tower and allowing the reaction to proceed for 1-5 hours until a substantial portion of the residual hydrogen peroxide has been consumed. It may be desirable to introduce the pulp to a mixer and add alkali to reactivate residual hydrogen peroxide by bringing the pulp to a pH of at least 9 between bleaching steps.