Many top-down architectures for instance segmentation achieve significant success when trained and tested on pre-defined closed-world taxonomy. However, when deployed in the open world, they exhibit notable bias towards seen classes and suffer from significant performance drop. In this work, we propose a novel approach for open world instance segmentation called bottom-Up and top-Down Open-world Segmentation (UDOS) that combines classical bottom-up segmentation algorithms within a top-down learning framework. UDOS first predicts parts of objects using a top-down network trained with weak supervision from bottom-up segmentations. The bottom-up segmentations are class-agnostic and do not overfit to specific taxonomies. The part-masks are then fed into affinity-based grouping and refinement modules to predict robust instance-level segmentations. UDOS enjoys both the speed and efficiency from the top-down architectures and the generalization ability to unseen categories from bottom-up supervision. We validate the strengths of UDOS on multiple cross-category as well as cross-dataset transfer tasks from 5 challenging datasets including MS-COCO, LVIS, ADE20k, UVO and OpenImages, achieving significant improvements over state-of-the-art across the board. Our code and models are available on our project page.