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Moving Object Detection (MOD) is a critical vision task for successfully achieving safe autonomous driving. Despite plausible results of deep learning methods, most existing approaches are only frame-based and may fail to reach reasonable performance when dealing with dynamic traffic participants. Recent advances in sensor technologies, especially the Event camera, can naturally complement the conventional camera approach to better model moving objects. However, event-based works often adopt a pre-defined time window for event representation, and simply integrate it to estimate image intensities from events, neglecting much of the rich temporal information from the available asynchronous events. Therefore, from a new perspective, we propose RENet, a novel RGB-Event fusion Network, that jointly exploits the two complementary modalities to achieve more robust MOD under challenging scenarios for autonomous driving. Specifically, we first design a temporal multi-scale aggregation module to fully leverage event frames from both the RGB exposure time and larger intervals. Then we introduce a bi-directional fusion module to attentively calibrate and fuse multi-modal features. To evaluate the performance of our network, we carefully select and annotate a sub-MOD dataset from the commonly used DSEC dataset. Extensive experiments demonstrate that our proposed method performs significantly better than the state-of-the-art RGB-Event fusion alternatives. The source code and dataset are publicly available at: https://github.com/ZZY-Zhou/RENet.