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Learning accurate object detectors often requires large-scale training data with precise object bounding boxes. However, labeling such data is expensive and time-consuming. As the crowd-sourcing labeling process and the ambiguities of the objects may raise noisy bounding box annotations, the object detectors will suffer from the degenerated training data. In this work, we aim to address the challenge of learning robust object detectors with inaccurate bounding boxes. Inspired by the fact that localization precision suffers significantly from inaccurate bounding boxes while classification accuracy is less affected, we propose leveraging classification as a guidance signal for refining localization results. Specifically, by treating an object as a bag of instances, we introduce an Object-Aware Multiple Instance Learning approach (OA-MIL), featured with object-aware instance selection and object-aware instance extension. The former aims to select accurate instances for training, instead of directly using inaccurate box annotations. The latter focuses on generating high-quality instances for selection. Extensive experiments on synthetic noisy datasets (i.e., noisy PASCAL VOC and MS-COCO) and a real noisy wheat head dataset demonstrate the effectiveness of our OA-MIL. Code is available at https://github.com/cxliu0/OA-MIL.