Scanpath prediction for 360° images aims to produce dynamic gaze behaviors based on the human visual perception mechanism. Most existing scanpath prediction methods for 360° images do not give a complete treatment of the time-dependency when predicting human scanpath, resulting in inferior performance and poor generalizability. In this paper, we present a scanpath prediction method for 360° images by designing a novel Deep Markov Model (DMM) architecture, namely ScanDMM. We propose a semantics-guided transition function to learn the nonlinear dynamics of time-dependent attentional landscape. Moreover, a state initialization strategy is proposed by considering the starting point of viewing, enabling the model to learn the dynamics with the correct “launcher”. We further demonstrate that our model achieves state-of-the-art performance on four 360° image databases, and exhibit its generalizability by presenting two applications of applying scanpath prediction models to other visual tasks - saliency detection and image quality assessment, expecting to provide profound insights into these fields.