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Designed for tracking user goals in dialogues, a dialogue state tracker is an essential component in a dialogue system. However, the research of dialogue state tracking has largely been limited to unimodality, in which slots and slot values are limited by knowledge domains (e.g. restaurant domain with slots of restaurant name and price range) and are defined by specific database schema. In this paper, we propose to extend the definition of dialogue state tracking to multimodality. Specifically, we introduce a novel dialogue state tracking task to track the information of visual objects that are mentioned in video-grounded dialogues. Each new dialogue utterance may introduce a new video segment, new visual objects, or new object attributes, and a state tracker is required to update these information slots accordingly. We created a new synthetic benchmark and designed a novel baseline, Video-Dialogue Transformer Network (VDTN), for this task. VDTN combines both object-level features and segment-level features and learns contextual dependencies between videos and dialogues to generate multimodal dialogue states. We optimized VDTN for a state generation task as well as a self-supervised video understanding task which recovers video segment or object representations. Finally, we trained VDTN to use the decoded states in a response prediction task. Together with comprehensive ablation and qualitative analysis, we discovered interesting insights towards building more capable multimodal dialogue systems.