Video object removal aims at erasing a target object in the entire video and filling holes with plausible contents, given an object mask in the first frame as input. Existing solutions mostly break down the task into (supervised) mask tracking and (self-supervised) video completion, and then separately tackle them with tailored designs. In this paper, we introduce a new setup, coined as unified video object removal, where mask tracking and completion are addressed within a unified framework. Despite introducing more challenges, the setup is promising for future practical usage. We embrace the observation that these two sub-tasks have strong inherent connections in terms of pixel-level temporal correspondence. Making full use of the connections could be beneficial considering the complexity of both algorithm and deployment. We propose a single network linking the two sub-tasks by inferring temporal correspondences across multiple frames, i.e., correspondences between valid-valid (V-V) pixel pairs for mask tracking and correspondences between valid-hole (V-H) pixel pairs for video completion. Thanks to the unified setup, the network can be learned end-to-end in a totally unsupervised fashion without any annotations. We demonstrate that our method can generate visually pleasing results and perform favorably against existing separate solutions in realistic test cases.