Three-dimensional object detection is one of the key tasks in autonomous driving. To reduce costs in practice, low-cost multi-view cameras for 3D object detection are proposed to replace the expansive LiDAR sensors. However, relying solely on cameras is difficult to achieve highly accurate and robust 3D object detection. An effective solution to this issue is combining multi-view cameras with the economical millimeter-wave radar sensor to achieve more reliable multi-modal 3D object detection. In this paper, we introduce RCBEVDet, a radar-camera fusion 3D object detection method in the bird's eye view (BEV). Specifically, we first design RadarBEVNet for radar BEV feature extraction. RadarBEVNet consists of a dual-stream radar backbone and a Radar Cross-Section (RCS) aware BEV encoder. In the dual-stream radar backbone, a point-based encoder and a transformer-based encoder are proposed to extract radar features, with an injection and extraction module to facilitate communication between the two encoders. The RCS-aware BEV encoder takes RCS as the object size prior to scattering the point feature in BEV. Besides, we present the Cross-Attention Multi-layer Fusion module to automatically align the multi-modal BEV feature from radar and camera with the deformable attention mechanism, and then fuse the feature with channel and spatial fusion layers. Experimental results show that RCBEVDet achieves new state-of-the-art radar-camera fusion results on nuScenes and view-of-delft (VoD) 3D object detection benchmarks. Furthermore, RCBEVDet achieves better 3D detection results than all real-time camera-only and radar-camera 3D object detectors with a faster inference speed at 21~28 FPS. The source code and models will be released to the public.