  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.
