Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Levy 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.