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With the continuous research on Deepfake forensics, recent studies have attempted to provide the fine-grained localization of forgeries, in addition to the coarse classification at the video-level. However, the detection and localization performance of existing Deepfake forensic methods still have plenty of room for further improvement. In this work, we propose a Spatial-Temporal Deepfake Detection and Localization (ST-DDL) network that simultaneously explores spatial and temporal features for detecting and localizing forged regions. Specifically, we design a new Anchor-Mesh Motion (AMM) algorithm to extract temporal (motion) features by modeling the precise geometric movements of the facial micro-expression. Compared with traditional motion extraction methods (e.g., optical flow) designed to simulate large-moving objects, our proposed AMM could better capture the small-displacement facial features. The temporal features and the spatial features are then fused in a Fusion Attention (FA) module based on a Transformer architecture for the eventual Deepfake forensic tasks. The superiority of our ST-DDL network is verified by experimental comparisons with several state-of-the-art competitors, in terms of both video- and pixel-level detection and localization performance. Furthermore, to impel the future development of Deepfake forensics, we build a public forgery dataset consisting of 6000 videos, with many new features such as using widely-used commercial software (e.g., After Effects) for the production, providing online social networks transmitted versions, and splicing multi-source videos. The source code and dataset are available at https://github.com/HighwayWu/ST-DDL.