McNeill (1985) claimed that speech and its accompanying gestures "share a common computational stage" that is located early in the speech production process. This position depends on two crucial arguments that we believe are unsound: first, that gestures "are synchronized with linguistic units in speech" and, second, that gestures "have semantic and pragmatic functions that parallel those of speech." Careful analysis of the relevant data indicated that although the processes underlying gestural production are globally autonomous, they are linked to speech production at more than just one computational stage. An account is offered that explicates these links.