Hyperspectral anomaly detection can separate sparse anomalies from the low-rank background component under an unsupervised behavior due to sufficient spectral information. Therefore, hyperspectral image anomaly detection technology has great application potential and value in public security and national defense. Currently, most existing models attempt to detect anomalous targets with a sparsity prior, without further considering the visual saliency of the targets themselves. To tackle this issue, this paper proposes a saliency-guided sparse low-rank tensor approximation model, called SSLR, to detect anomalous targets from hyperspectral remote sensing images in an unsupervised manner. Specifically, we first explore the saliency information of each pixel for regularizing the sparse anomaly matrix. We then suggest a three-directional tensor nuclear norm to obtain a low-rank background to characterize the background component. We solve the SSLR optimization problem by an efficient alternating direction method of multipliers framework. Experiments conducted on benchmark hyperspectral datasets demonstrate that the proposed SSLR outperforms some state-of-the-art anomaly detection methods.