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In this paper, we address the problem of estimating dense depth from a sequence of images using deep neural networks. Specifically, we employ a dense-optical-flow network to compute correspondences and then triangulate the point cloud to obtain an initial depth map.Parts of the point cloud, however, may be less accurate than others due to lack of common observations or small parallax. To further increase the triangulation accuracy, we introduce a depth-refinement network (DRN) that optimizes the initial depth map based on the image's contextual cues. In particular, the DRN contains an iterative refinement module (IRM) that improves the depth accuracy over iterations by refining the deep features. Lastly, the DRN also predicts the uncertainty in the refined depths, which is desirable in applications such as measurement selection for scene reconstruction. We show experimentally that our algorithm outperforms state-of-the-art approaches in terms of depth accuracy, and verify that our predicted uncertainty is highly correlated to the actual depth error.