Even though language allows us to say exactly what we mean, we often use language to say things indirectly, in a way that depends on the specific communicative context. For example, we can use an apparently straightforward sentence like "It is hard to give a good presentation" to convey deeper meanings, like "Your talk was a mess!" One of the big puzzles in language science is how listeners work out what speakers really mean, which is a skill absolutely central to communication. However, most neuroimaging studies of language comprehension have focused on the arguably much simpler, context-independent process of understanding direct utterances. To examine the neural systems involved in getting at contextually constrained indirect meaning, we used functional magnetic resonance imaging as people listened to indirect replies in spoken dialog. Relative to direct control utterances, indirect replies engaged dorsomedial prefrontal cortex, right temporo-parietal junction and insula, as well as bilateral inferior frontal gyrus and right medial temporal gyrus. This suggests that listeners take the speaker's perspective on both cognitive (theory of mind) and affective (empathy-like) levels. In line with classic pragmatic theories, our results also indicate that currently popular "simulationist" accounts of language comprehension fail to explain how listeners understand the speaker's intended message.