After years of promise conversational systems such as social chatbots, automated helpdesks, and chat-based search are hitting the mainstream. These systems are increasingly competent at the fundamentals of informational or social conversation: they are finally tracking topics across multiple turns, “understanding” many requests, and forming grammatical and coherent responses. So what else is there to do? Conversation without computers, of course, has been well-studied for decades. Research has analysed linguistic phenomena such as structure and semantics but also paralinguistic features such as tone, body language, and participants’ physiological states. This work gives us some strong hints where we should focus next, as we try to build conversational agents which are fluent as well as grammatical, pleasant as well as correct. In this talk we’ll take a quick tour through some past work, illustrated with examples, and discuss where things could (or should) be headed.