This paper describes an efficient and effective design of Robust Spatio-Temporal Prediction based on Student's t distribution, namely, St-RSTP, to provide estimations based on observations over spatio-temporal neighbors. The proposed St-RSTP is more resilient to outliers or other small departures from model assumptions than its ancestor, the Spatio-Temporal Random Effects (STRE) model. STRE is a state-of-the-art statistical model with linear order complexity for large scale processing. However, it assumes Gaussian observations, which has the well-known limitation of non-robustness. In our St-RSTP design, the measurement error follows Student's t distribution, instead of a traditional Gaussian distribution. This design reduces the influence of outliers, improves prediction quality, and keeps the problem analytically intractable. We propose a novel approximate inference approach, which approximates the model into the form that separates the high dimensional latent variables into groups, and then estimates the posterior distributions of different groups of variables separately in the framework of Expectation Propagation. As a good property, our approximate approach degeneralizes to the standard STRE based prediction, when the degree of freedom of the Student's t distribution is set to infinite. Extensive experimental evaluations based on both simulation and real-life data sets demonstrated the robustness and the efficiency of our Student-t prediction model. The proposed approach provides critical functionality for stochastic processes on spatio-temporal data.