  Gaussian random fields with Mat\'ern covariance functions are popular models
in spatial statistics and machine learning. In this work, we develop a
spatio-temporal extension of the Gaussian Mat\'ern fields formulated as
solutions to a stochastic partial differential equation. The spatially
stationary subset of the models have marginal spatial Mat\'ern covariances, and
the model also extends to Whittle-Mat\'ern fields on curved manifolds, and to
more general non-stationary fields. In addition to the parameters of the
spatial dependence (variance, smoothness, and practical correlation range) it
additionally has parameters controlling the practical correlation range in
time, the smoothness in time, and the type of non-separability of the
spatio-temporal covariance. Through the separability parameter, the model also
allows for separable covariance functions. We provide a sparse representation
based on a finite element approximation, that is well suited for statistical
inference and which is implemented in the R-INLA software. The flexibility of
the model is illustrated in an application to spatio-temporal modeling of
global temperature data.
