Random fields (RFs) and Gaussian processes (GPs), also known as kriging, are two established categories of models designed for spatial and spatiotemporal data. RFs already have established use in the statistical modeling of microstructured materials , while GPs are invented in mining engineering and they are a classic example of surrogate models. We discuss here three diverse examples from materials data science that indicate their broad applicability and utility. (i) In metal additive manufacturing, Saunders et al. combined three GPs with distinct characteristics to model the pairwise relationships between materials microstructure, melt pool geometry, and mechanical property obtained from multiphysics simulation, all of which are also time-dependent. (ii) In photoemission spectroscopy, Xian and Stimper et al. constructed a Markov RF with nearest-neighbor interaction and transformed the band dispersion reconstruction problem into a classification problem. The coordinates in their problem are the two momenta and energy of photoelectrons, the use of pre-computed energy bands from electronic structure theory provides an effective initialization. (iii) In combinatorial materials screening, Kusne et al. constructed a GP in the chemical compositional space to guide the search for the optimal stoichiometry within a family of tertiary phase-change materials. Their algorithm was integrated into a synchrotron beamline and may be run in a closed loop driven by active learning.