In both insect and vertebrate olfactory systems only two synapses separate the sensory periphery from brain areas required for memory formation and the organisation of behaviour. In the Drosophila olfactory system, which is anatomically very similar to its vertebrate counterpart, there has been substantial recent progress in understanding the flow of information from experiments using molecular genetic, electrophysiological and optical imaging techniques. In this review, we shall focus on how olfactory information is processed and transformed in order to extract behaviourally relevant information. We follow the progress from olfactory receptor neurons, through the first processing area, the antennal lobe, to higher olfactory centres. We address both the underlying anatomy and mechanisms that govern the transformation of neural activity. We emphasise our emerging understanding of how different elementary computations, including signal averaging, gain control, decorrelation and integration, may be mapped onto different circuit elements.