Depth sensing is an important problem for 3D vision-based robotics. Yet, a real-world active stereo or ToF depth camera often produces noisy and incomplete depth which bottlenecks robot performances. In this work, we propose D3RoMa, a learning-based depth estimation framework on stereo image pairs that predicts clean and accurate depth in diverse indoor scenes, even in the most challenging scenarios with translucent or specular surfaces where classical depth sensing completely fails. Key to our method is that we unify depth estimation and restoration into an image-to-image translation problem by predicting the disparity map with a denoising diffusion probabilistic model. At inference time, we further incorporated a left-right consistency constraint as classifier guidance to the diffusion process. Our framework combines recently advanced learning-based approaches and geometric constraints from traditional stereo vision. For model training, we create a large scene-level synthetic dataset with diverse transparent and specular objects to compensate for existing tabletop datasets. The trained model can be directly applied to real-world in-the-wild scenes and achieve state-of-the-art performance in multiple public depth estimation benchmarks. Further experiments in real environments show that accurate depth prediction significantly improves robotic manipulation in various scenarios.