Entanglement entropy first arose from attempts to understand the entropy of black holes, and is believed to play a crucial role in a complete description of quantum gravity. This thesis explores some proposed connections between entanglement entropy and the geometry of spacetime. One such connection is the ability to derive gravitational field equations from entanglement identities. I will discuss a specific derivation of the Einstein equation from an equilibrium condition satisfied by entanglement entropy, and explore a subtlety in the construction when the matter fields are not conformally invariant. As a further generalization, I extend the argument to include higher curvature theories of gravity, whose consideration is necessitated by the presence of subleading divergences in the entanglement entropy beyond the area law.

A deeper issue in this construction, as well as in more general considerations identifying black hole entropy with entanglement entropy, is that the entropy is ambiguous for gauge fields and gravitons. The ambiguity stems from how one handles edge modes at the entangling surface, which parameterize the gauge transformations that are broken by the presence of the boundary. The final part of this thesis is devoted to identifying the edge modes in arbitrary diffeomorphism-invariant theories. Edge modes are conjectured to provide a statistical description of the black hole entropy, and this work takes some initial steps toward checking this conjecture in higher curvature theories.