  We propose a novel class of tempo-spatial Ornstein-Uhlenbeck processes as
solutions to L\'evy-driven Volterra equations with additive noise and
multiplicative drift. After formulating conditions for the existence and
uniqueness of solutions, we derive an explicit solution formula and discuss
distributional properties such as stationarity, second-order structure and
short versus long memory. Furthermore, we analyze in detail the path properties
of the solution process. In particular, we introduce different notions of
c\`adl\`ag paths in space and time and establish conditions for the existence
of versions with these regularity properties. The theoretical results are
accompanied by illustrative examples.
