The core niche for language use is in verbal interaction, involving the rapid exchange of turns at talking. This paper reviews the extensive literature about this system, adding new statistical analyses of behavioural data where they have been missing, demonstrating that turn-taking has the systematic properties originally noted by Sacks, Schegloff and Jefferson (1974; hereafter SSJ). This system poses some significant puzzles for current theories of language processing: the gaps between turns are short (of the order of 200 ms), but the latencies involved in language production are much longer (over 600 ms). This seems to imply that participants in conversation must predict (or ‘project’ as SSJ have it) the end of the current speaker’s turn in order to prepare their response in advance. This in turn implies some overlap between production and comprehension despite their use of common processing resources. Collecting together what is known behaviourally and experimentally about the system, the space for systematic explanations of language processing for conversation can be significantly narrowed, and we sketch some first model of the mental processes involved for the participant preparing to speak next.