Existing work on object detection often relies on a single form of annotation: the model is trained using either accurate yet costly bounding boxes or cheaper but less expressive image-level tags. However, real-world annotations are often diverse in form, which challenges these existing works. In this paper, we present UFO2, a unified object detection framework that can handle different forms of supervision simultaneously. Specifically, UFO2 incorporates strong supervision (e.g., boxes), various forms of partial supervision (e.g., class tags, points, and scribbles), and unlabeled data. Through rigorous evaluations, we demonstrate that each form of label can be utilized to either train a model from scratch or to further improve a pre-trained model. We also use UFO2 to investigate budget-aware omni-supervised learning, i.e., various annotation policies are studied under a fixed annotation budget: we show that competitive performance needs no strong labels for all data. Finally, we demonstrate the generalization of UFO2, detecting more than 1,000 different objects without bounding box annotations.