Segmenting object parts such as cup handles and animal bodies is important in many real-world applications but requires more annotation effort. The largest dataset nowadays contains merely two hundred object categories, implying the difficulty to scale up part segmentation to an unconstrained setting. To address this, we propose to explore a seemingly simplified but empirically useful and scalable task, class-agnostic part segmentation. In this problem, we disregard the part class labels in training and instead treat all of them as a single part class. We argue and demonstrate that models trained without part classes can better localize parts and segment them on objects unseen in training. We then present two further improvements. First, we propose to make the model object-aware, leveraging the fact that parts are “compositions” whose extents are bounded by objects, whose appearances are by nature not independent but bundled. Second, we introduce a novel approach to improve part segmentation on unseen objects, inspired by an interesting finding --- for unseen objects, the pixel-wise features extracted by the model often reveal high-quality part segments. To this end, we propose a novel self-supervised procedure that iterates between pixel clustering and supervised contrastive learning that pulls pixels closer or pushes them away. Via extensive experiments on PartImageNet and Pascal-Part, we show notable and consistent gains by our approach, essentially a critical step towards open-world part segmentation.