Monocular 3D object detection is an important yet challenging task in autonomous driving. Some existing methods leverage depth information from an off-the-shelf depth estimator to assist 3D detection, but suffer from the additional computational burden and only achieve sub-optimal performance led by inaccurate depth priors. To alleviate this, we propose MonoDTR, a novel end-to-end depth-aware transformer network for monocular 3D object detection. It mainly consists of two components: (1) the Depth-Aware Feature Enhancement (DFE) module that implicitly learns depth-aware features with auxiliary supervision without requiring extra computation, and (2) the Depth-Aware Transformer (DTR) module that globally integrates context- and depth-aware features. Moreover, different from conventional pixel-wise positional encodings, we introduce a novel depth positional encoding (DPE) to inject depth positional hints into transformers. Our proposed depth-aware modules can be easily plugged into existing image-only monocular 3D object detectors to improve the performance. Extensive experiments on the KITTI dataset demonstrate that our approach outperforms previous state-of-the-art monocular-based methods and achieves real-time detection.