  We review the probabilistic properties of Ornstein-Uhlenbeck processes in
Hilbert spaces driven by L\'{e}vy processes. The emphasis is on the different
contexts in which these processes arise, such as stochastic partial
differential equations, continuous-state branching processes, generalised
Mehler semigroups and operator self-decomposable distributions. We also examine
generalisations to the case where the driving noise is cylindrical.
