There is growing interest in robots that are designed specifically to interact with people and which respond to voice commands. Very little attention has been paid, however, to the kind of verbal interaction that is possible or desirable with robots. This article presents recent work in multimodal interfaces that addresses this question. It proposes a new form of robot-user interface, namely a collaborative conversational interface. This article explains what collaborative conversational interfaces are, argues for their application in robots, and presents strategies for designing good conversational interfaces. The emphasis throughout the paper is on robots, such as rehabilitation robots, that are used within healthcare. The paper concludes with a discussion of the particular challenges faced in designing conversational interfaces for robots.