Self-supervised monocular depth estimation has achieved notable success under daytime conditions. However, its performance deteriorates markedly at night due to low visibility and varying illumination, e.g., insufficient light causes textureless areas, and moving objects bring blurry regions. To this end, we propose a self-supervised framework named DASP that leverages spatiotemporal priors for nighttime depth estimation. Specifically, DASP consists of an adversarial branch for extracting spatiotemporal priors and a self-supervised branch for learning. In the adversarial branch, we first design an adversarial network where the discriminator is composed of four devised spatiotemporal priors learning blocks (SPLB) to exploit the daytime priors. In particular, the SPLB contains a spatial-based temporal learning module (STLM) that uses orthogonal differencing to extract motion-related variations along the time axis and an axial spatial learning module (ASLM) that adopts local asymmetric convolutions with global axial attention to capture the multiscale structural information. By combining STLM and ASLM, our model can acquire sufficient spatiotemporal features to restore textureless areas and estimate the blurry regions caused by dynamic objects. In the self-supervised branch, we propose a 3D consistency projection loss to bilaterally project the target frame and source frame into a shared 3D space, and calculate the 3D discrepancy between the two projected frames as a loss to optimize the 3D structural consistency and daytime priors. Extensive experiments on the Oxford RobotCar and nuScenes datasets demonstrate that our approach achieves state-of-the-art performance for nighttime depth estimation. Ablation studies further validate the effectiveness of each component.