  Gaussian processes and random fields have a long history, covering multiple
approaches to representing spatial and spatio-temporal 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 Mat\'ern
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.
