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Monocular 3D object detection (Mono3D) has achieved unprecedented success with the advent of deep learning techniques and emerging large-scale autonomous driving datasets. However, drastic performance degradation remains an unwell-studied challenge for practical cross-domain deployment as the lack of labels on the target domain. In this paper, we first comprehensively investigate the significant underlying factor of the domain gap in Mono3D, where the critical observation is a depth-shift issue caused by the geometric misalignment of domains. Then, we propose STMono3D, a new self-teaching framework for unsupervised domain adaptation on Mono3D. To mitigate the depth-shift, we introduce the geometry-aligned multi-scale training strategy to disentangle the camera parameters and guarantee the geometry consistency of domains. Based on this, we develop a teacher-student paradigm to generate adaptive pseudo labels on the target domain. Benefiting from the end-to-end framework that provides richer information of the pseudo labels, we propose the quality-aware supervision strategy to take instance-level pseudo confidences into account and improve the effectiveness of the target-domain training process. Moreover, the positive focusing training strategy and dynamic threshold are proposed to handle tremendous FN and FP pseudo samples. STMono3D achieves remarkable performance on all evaluated datasets and even surpasses fully supervised results on the KITTI 3D object detection dataset. To the best of our knowledge, this is the first study to explore effective UDA methods for Mono3D.