RGB-D saliency detection has attracted increasing attention, due to its effectiveness and the fact that depth cues can now be conveniently captured. Existing works often focus on learning a shared representation through various fusion strategies, with few methods explicitly considering how to preserve modality-specific characteristics. In this paper, taking a new perspective, we propose a specificity-preserving network (SP-Net) for RGB-D saliency detection, which benefits saliency detection performance by exploring both the shared information and modality-specific properties (e.g., specificity). Specifically, two modality-specific networks and a shared learning network are adopted to generate individual and shared saliency maps. A cross-enhanced integration module (CIM) is proposed to fuse cross-modal features in the shared learning network, which are then propagated to the next layer for integrating cross-level information. Besides, we propose a multi-modal feature aggregation (MFA) module to integrate the modality-specific features from each individual decoder into the shared decoder, which can provide rich complementary multi-modal information to boost the saliency detection performance. Further, a skip connection is used to combine hierarchical features between the encoder and decoder layers. Experiments on six benchmark datasets demonstrate that our SP-Net outperforms other state-of-the-art methods. Code is available at: https://github.com/taozh2017/SPNet.