Image landmark detection aims to automatically identify the locations of predefined fiducial points. Despite recent success in this field, higher-ordered structural modeling to capture implicit or explicit relationships among anatomical landmarks has not been adequately exploited. In this work, we present a new topology-adapting deep graph learning approach for accurate anatomical facial and medical (e.g., hand, pelvis) landmark detection. The proposed method constructs graph signals leveraging both local image features and global shape features. The adaptive graph topology naturally explores and lands on task-specific structures which are learned end-to-end with two Graph Convolutional Networks (GCNs). Extensive experiments are conducted on three public facial image datasets (WFLW, 300W, and COFW-68) as well as three real-world X-ray medical datasets (Cephalometric (public), Hand and Pelvis). Quantitative results comparing with the previous state-of-the-art approaches across all studied datasets indicating the superior performance in both robustness and accuracy. Qualitative visualizations of the learned graph topologies demonstrate a physically plausible connectivity laying behind the landmarks.