  The entanglement entropy (EE) can measure the entanglement between a spatial
subregion and its complement, which provides key information about quantum
states. Here, rather than focusing on specific regions, we study how the
entanglement entropy changes with small deformations of the entangling surface.
This leads to the notion of entanglement susceptibilities. These relate the
variation of the EE to the geometric variation of the subregion. We determine
the form of the leading entanglement susceptibilities for a large class of
scale invariant states, such as groundstates of conformal field theories, and
systems with Lifshitz scaling, which includes fixed points governed by
disorder. We then use the susceptibilities to derive the universal
contributions that arise due to non-smooth features in the entangling surface:
corners in 2d, as well as cones and trihedral vertices in 3d. We finally
discuss the generalization to Renyi entropies.
