At present, most saliency detection methods are based on fully convolutional neural networks (FCNs). However, FCNs usually blur the edges of salient objects. Due to that, the multiple convolution and pooling operations of the FCNs will limit the spatial resolution of the feature maps. To alleviate this issue and obtain accurate edges, we propose a hierarchical edge refinement network (HERNet) for accurate saliency detection. In detail, the HERNet is mainly composed of a saliency prediction network and an edge preserving network. Firstly, the saliency prediction network is used to roughly detect the regions of salient objects and is based on a modified U-Net structure. Then, the edge preserving network is used to accurately detect the edges of salient objects, and this network is mainly composed of the atrous spatial pyramid pooling (ASPP) module. Different from the previous indiscriminate supervision strategy, we adopt a new one-to-one hierarchical supervision strategy to supervise the different outputs of the entire network. Experimental results on five traditional benchmark datasets demonstrate that the proposed HERNet performs well when compared with the state-of-the-art methods.