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Growing interests in RGB-D salient object detection (RGB-D SOD) have been witnessed in recent years, owing partly to the popularity of depth sensors and the rapid progress of deep learning techniques. Unfortunately, existing RGB-D SOD methods typically demand large quantity of training images being thoroughly annotated at pixel-level. The laborious and time-consuming manual annotation has become a real bottleneck in various practical scenarios. On the other hand, current unsupervised RGB-D SOD methods still heavily rely on handcrafted feature representations. This inspires us to propose in this paper a deep unsupervised RGB-D saliency detection approach, which requires no manual pixel-level annotation during training. It is realized by two key ingredients in our training pipeline. First, a depth-disentangled saliency update (DSU) framework is designed to automatically produce pseudo-labels with iterative follow-up refinements, which provides more trustworthy supervision signals for training the saliency network. Second, an attentive training strategy is introduced to tackle the issue of noisy pseudo-labels, by properly re-weighting to highlight the more reliable pseudo-labels. Extensive experiments demonstrate the superior efficiency and effectiveness of our approach in tackling the challenging unsupervised RGB-D SOD scenarios. Moreover, our approach can also be adapted to work in fully-supervised situation. Empirical studies show the incorporation of our approach gives rise to notably performance improvement in existing supervised RGB-D SOD models.