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It would be a technological feat to be able to create a system that can hold a meaningful conversation with humans about what they watch. A setup toward that goal is presented as a video dialog task, where the system is asked to generate natural utterances in response to a question in an ongoing dialog. The task poses great visual, linguistic, and reasoning challenges that cannot be easily overcome without an appropriate representation scheme over video and dialog that supports high-level reasoning. To tackle these challenges we present a new object-centric framework for video dialog that supports neural reasoning dubbed COST - which stands for Conversation about Objects in Space-Time. Here dynamic space-time visual content in videos is first parsed into object trajectories. Given this video abstraction, COST maintains and tracks object-associated dialog states, which are updated upon receiving new questions. Object interactions are dynamically and conditionally inferred for each question, and these serve as the basis for relational reasoning among them. COST also maintains a history of previous answers, and this allows retrieval of relevant object-centric information to enrich the answer forming process. Language production then proceeds in a step-wise manner, taking into the context of the current utterance, the existing dialog, the current question. We evaluate COST on the DSTC7 and DSTC8 benchmarks, demonstrating its competitiveness against state-of-the-arts.