The invention provides liquid bleaching compositions comprising multiple partial compositions held separately from each other in a multiple compartment container. One partial composition comprises a peroxygen bleach compound, preferably hydrogen peroxide. One partial composition (which may be the same as the one containing the peroxygen compound) comprises an imine or oxaziridine bleach activator compound. One partial composition, which is different from the one containing the bleach activator, comprises an alkaline compound in amount sufficient to raise the pH of the total composition to a value at which the bleach system is effective. The bleach activator compound is preferably a sulfonimine or a quaternary imine salt. The number of partial compositions and of the compartments in the container is preferably 2.