Much of materials science research is a “small data” problem – an experimental or computational result may contain many data points, perhaps an entire space or time field of outputs, but there might only be several dozens of these outputs. So materials scientists cannot rely on techniques of analysis that are data hungry, requiring hundreds of thousands of training datasets. We discuss recent ideas in employing Gaussian process surrogate models, a methodology that provides good predictive capability based on relatively modest data needs, and which comes with objective measures of credibility in those predictions. In particular, we discuss how Gaussian processes are being extended to handle fields of inputs and/or outputs.