Entanglement entropy of quantum fields is a topic of growing importance in several areas such as quantum gravity, condensed matter physics, and quantum information. In this talk I will review a formulation of entanglement entropy for Gaussian theories, in terms of the spacetime two-point correlation function. One of the attractive features of this formulation is that it makes it possible to use a UV cutoff that is spacetime in nature. I will discuss applications of this formulation in settings in both continuum spacetimes and discrete causal sets, as well as extensions to non-Gaussian and interacting theories.