We present recent results regarding escape rates and conditionally invariant measures for a periodic Lorentz gas with holes. We then derive a variational principle connecting the escape rate to the pressure on the survivor set, the set of points which never enters the hole. This relation generalizes to a broad class of systems with holes and requires only weak assumptions on the size and boundary of the hole. When the underlying dynamical system is smooth (before the introduction of the hole) the variational principle allows us to determine how the escape rate changes as we vary the size and position of the hole. This is joint work with Paul Wright and Lai-Sang Young.