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In this work, we propose a 3D fully convolutional architecture for video saliency prediction that employs hierarchical supervision on intermediate maps (referred to as conspicuity maps) generated using features extracted at different abstraction levels. We provide the base hierarchical learning mechanism with two techniques for domain adaptation and domain-specific learning. For the former, we encourage the model to unsupervisedly learn hierarchical general features using gradient reversal at multiple scales, to enhance generalization capabilities on datasets for which no annotations are provided during training. As for domain specialization, we employ domain-specific operations (namely, priors, smoothing and batch normalization) by specializing the learned features on individual datasets in order to maximize performance. The results of our experiments show that the proposed model yields state-of-the-art accuracy on supervised saliency prediction. When the base hierarchical model is empowered with domain-specific modules, performance improves, outperforming state-of-the-art models on three out of five metrics on the DHF1K benchmark and reaching the second-best results on the other two. When, instead, we test it in an unsupervised domain adaptation setting, by enabling hierarchical gradient reversal layers, we obtain performance comparable to supervised state-of-the-art.