Introducing a small hole into the phase space of an ergodic dynamical system causes almost every trajectory to eventually escape. Despite this, such systems can have rich dynamics. For dispersing billiards with holes, we construct physically relevant invariant and conditionally invariant measures with properties analogous to those of SRB measures for closed systems. We also prove that the conditionally invariant measures converge to the smooth invariant measure in the small hole limit and establish a variational principle for the open system. This is joint work with Lai-Sang Young and Paul Wright.