Although the language we encounter is typically embedded in rich discourse contexts, existing models of sentence processing focus largely on phenomena that occur sentence internally. Here we analyze a video corpus of child-caregiver interactions with the aim of characterizing how discourse structure is reflected in child-directed speech and in children's and caregivers' behavior. We use topic continuity as a measure of discourse structure, examining how caregivers introduce and discuss objects across sentences. We develop a variant on a Hidden Markov Model to identify discourses, using speakers' intended referent and the time delays between utterances. Using the discourses found by this model, we analyze how the lexical, syntactic, and social properties of caregiver-child interaction change over the course of a sequence of topically-related utterances. Our findings suggest that cues used to signal topicality in adult discourse are also available in child-directed speech and that children's responses reflect joint attention in communication.