We will discuss the possible closures of geodesic planes in a hyperbolic 3-manifold M. When M has finite volume Shah and Ratner (independently) showed that a strong rigidity phenomenon holds, and in particular such closures are always properly immersed submanifolds of M with finite area. We show that a similar rigidity phenomenon holds for a class of infinite volume manifolds. This is based on joint works with C. McMullen and H. Oh.