Reactive dyes of the formula I where a is 1 or 2, b is 0 or 1, Y is vinyl or a radical of the formula C2H4Q, where Q is an alkali-detachable group, L3 is a direct bond or a bridge member of the formula CO—NH—M1, where M1 is C2-C6-alkylene with or without interruption by 1 or 2 unadjacent oxygen atoms, imino or C1-C4-alkylimino groups, A1 is hydroxysulfonyl or a radical of the formula SO2Y, A2 is hydrogen, hydroxysulfonyl, methoxy, chlorine, bromine or carboxyl, W is either in case 1) the radical of a coupling component, of a monoazo dye or additionally, when b=0, of a disazo dye, which may each bear further fiber-reactive groups, or  in case 2) the radical of a chromophore which optionally has further reactive groups, L1 and L2 are each a bridge member, are useful for hair dyeing and for use in cosmetic preparations for dyeing hair.