Abstract
Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatiotemporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Matern covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space-time fields on arbitrary manifolds, and practical computational considerations. (c) 2022 Elsevier B.V. All rights reserved.