In this talk we 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 very strong rigidity phenomenon holds, and in particular such closures are always properly immersed submanifolds of M with finite area. Manifolds with infinite volume, however, are far less understood and are the main subject of this talk. This is based on a joint ongoing work with C. McMullen and H. Oh.