I argue here that functional neuroimaging data--which I restrict to the haemodynamic techniques of fMRI and PET--can inform psychological theorizing, provided one assumes a "systematic" function-structure mapping in the brain. In this case, imaging data simply comprise another dependent variable, along with behavioural data, that can be used to test competing theories. In particular, I distinguish two types of inference: function-to-structure deduction and structure-to-function induction. With the former inference, a qualitatively different pattern of activity over the brain under two experimental conditions implies at least one different function associated with changes in the independent variable. With the second type of inference, activity of the same brain region(s) under two conditions implies a common function, possibly not predicted a priori. I illustrate these inferences with imaging studies of recognition memory, short-term memory, and repetition priming. I then consider in greater detail what is meant by a "systematic" function-structure mapping and argue that, particularly for structure-to-function induction, this entails a one-to-one mapping between functional and structural units, although the structural unit may be a network of interacting regions and care must be taken over the appropriate level of functional/structural abstraction. Nonetheless, the assumption of a systematic function-structure mapping is a "working hypothesis" that, in common with other scientific fields, cannot be proved on independent grounds and is probably best evaluated by the success of the enterprise as a whole. I also consider statistical issues such as the definition of a qualitative difference and methodological issues such as the relationship between imaging and behavioural data. I finish by reviewing various objections to neuroimaging, including neophrenology, functionalism, and equipotentiality, and by observing some criticisms of current practice in the imaging literature.