Finite-dimensional diffusions have been successfully used as tractable approximations to gain insight into a large class of queueing systems. We show that certain classes of queueing systems lead naturally to infinite-dimensional diffusion approximations. For the particular model of many-server queues, we show that the limit state process converges to the solution of a stochastic partial differential equation subject to an unusual boundary condition that involves a coupled Ito equation. We also establish uniqueness of the stationary distribution of the pair of processes, and describe its implications for the original queueing system. This talk is based on joint works with Mohammadreza Aghajani and Haya Kaspi.