In the causal set approach to quantum gravity, the spacetime continuum is replaced by a fundamentally discrete graph, which is a locally finite partially ordered set. Studying quantum fields on such a discrete background can shed light on questions like black hole entropy and Hawking evaporation. I will describe how to construct the retarded scalar Green's functions on causal sets approximated by various spacetimes, including de Sitter and anti de Sitter spacetimes. As in the continuum, the massive Green's function can be obtained from the massless one, and hence the main task is to first identify the massless Green's function on the causal set. I will show how this can be done in 2, 3 and 4 dimensions. This can be used to extend Sorkin and Yazdi's calculation of entanglement entropy in 2 dimensions to more physically interesting spacetimes. I will end by describing some recent work in this direction.