Functional magnetic resonance imaging (fMRI) is now widely used in cognitive neuroscience to look for changes in neural activity that correlate with particular cognitive processes. But what does fMRI tell us? Until we know exactly what the fMRI signal represents, we will not know how to interpret the results of these studies. It is generally assumed that the fMRI signal is roughly proportional to a measure of local neural activity, averaged over several millimetres and several seconds — the linear transform model of the fMRI signal. The relationship between the signal and the underlying neural activity depends on the fMRI acquisition technique, the behavioural and stimulation protocol, and how neural activity is measured and quantified (for example, by firing rate, local field potential or synchronous firing). The linear transform model predicts that the fMRI response should sum over time. For example, the fMRI response to a 12-s stimulus should be the same as that to two 6-s stimuli. The frequently observed failure of this prediction could be due to a ceiling effect, or to the disproportionately large responses to very short stimuli that are seen in primary sensory cortex. It should be possible to predict the fMRI response from the underlying neural activity. Studies that have compared neural activity and fMRI responses indirectly have supported this relationship. A direct comparison of simultaneously recorded fMRI responses and neuronal signals in monkey visual cortex found that local field potentials predicted fMRI responses better than multi-unit activity, but that both were variable in accuracy. The linear transform model also predicts that fMRI responses and neural activity should be colocalized. There is increasing evidence that this is the case, although the analysis methods used influence the localization of the fMRI signal. It will be important to improve our understanding of the relationships between neural activity, blood flow and metabolism, and fMRI signals, if we are correctly to interpret fMRI studies. It will also be necessary to optimize protocols and analysis techniques if this powerful tool is to live up to its potential.