While traditional approaches to manipulation planning assume known object templates, recent approaches to "category-level manipulation" aim to manipulate a category of objects with potentially unknown instances and large intra-category shape variation. In this paper we explore an object representation to enable precise category-level manipulation, capturing a notion of the object configuration and extent, while being generalizable to novel instances. Building on our previous work, kPAM ¹ , we combine semantic keypoints with dense geometry (a point cloud or mesh) as the interface between the perception module and motion planner. Leveraging advances in learning-based keypoint detection and shape completion, both dense geometry and keypoints can be perceived from raw sensor input. Using the proposed hybrid object representation, we formulate the manipulation task as a motion planning problem which encodes both the object target configuration and physical feasibility for a category of objects. In this way, many existing manipulation planners can be generalized to categories of objects, and the resulting perception-to-action manipulation pipeline is robust to large intra-category shape variation. Extensive hardware experiments demonstrate our pipeline can produce robot trajectories that accomplish tasks with never-before-seen objects. The video demo is available on this link: https://sites.google.com/view/generalizable-manipulation.