Why are animal displays so complex? In contexts ranging from courtship and mating to parent-offspring communication to predator deterrence, biological signals often involve a number of different visual, auditory and/or olfactory components. Previous models of communication have tended to ignore this complexity, assuming that only one kind of display is available. Here, a new game-theoretical model of signalling is described, in which signallers may use more than one display to advertise their qualities. Additional displays may serve to enhance the accuracy with which receivers assess a single quality (the `backup signal' hypothesis), or to provide information about different qualities (the `multiple message' hypothesis). Multiple signals are shown to be stable, even when multiple receiver preferences entail significant costs, provided that signalling costs are strongly accelerating. In such cases, signallers bias their investment towards more efficient forms of signal, but not to the exclusion of other display types. When costs are not strongly accelerating, by contrast, individual signallers employ only a single display at equilibrium. If different signals provide information about different qualities, however, then the equilibrium may feature alternative signalling strategies, with signallers who excel in one quality employing one kind of display, and those who excel in another quality employing another kind.