This brief introduces a class of problems and models for the prediction of the scalar field of interest from noisy observations collected by mobile sensor networks. It also addresses the problem of optimal coordination of robotic sensors to maximize prediction quality while considering communication and mobility constraints, either in a centralized or distributed manner. To solve these problems, fully Bayesian approaches are adopted, allowing various sources of uncertainty to be integrated into an inferential framework that effectively captures all aspects of variability involved. The fully Bayesian approach also enables the automatic selection of the most appropriate values for additional model parameters based on data, achieving optimal inference and prediction for the underlying scalar field. In particular, spatio-temporal Gaussian process regression is formulated for robotic sensors.