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We present a robust and accurate depth refinement system, named GeoRefine, for geometrically-consistent dense mapping from monocular sequences. GeoRefine consists of three modules: a hybrid SLAM module using learning-based priors, an online depth refinement module leveraging self-supervision, and a global mapping module via TSDF fusion. The proposed system is online by design and achieves great robustness and accuracy via: (i) a robustified hybrid SLAM that incorporates learning-based optical flow and/or depth; (ii) self-supervised losses that leverage SLAM outputs and enforce long-term geometric consistency; (iii) careful system design that avoids degenerate cases in online depth refinement. We extensively evaluate GeoRefine on multiple public datasets and reach as low as 5% absolute relative depth errors.