  We introduce L\'evy-driven causal CARMA random fields on $\mathbb{R}^d$,
extending the class of CARMA processes. The definition is based on a system of
stochastic partial differential equations which generalize the classical
state-space representation of CARMA processes. The resulting CARMA model
differs fundamentally from the isotropic CARMA random field of Brockwell and
Matsuda. We show existence of the model under mild assumptions and examine some
of its features including the second-order structure and path properties. In
particular, we investigate the sampling behavior and formulate conditions for
the causal CARMA random field to be an ARMA random field when sampled on an
equidistant lattice.
