An entirely new kind of band insulator was discovered recently. These new electronic states - called "topological insulators" - are fundamentally different from standard band insulators. They are distinguished by the fact that their edges (in the 2D case) or surfaces (in the 3D case) support gapless transport which is extremely robust. In the two dimensional case, topological insulators can be thought of as time reversal invariant analogues of integer quantum Hall states. This analogy is intriguing since integer quantum Hall states are a special case of the far richer class of fractional quantum Hall states. It is natural to wonder: can topological insulators also be generalized? In this talk, I will investigate this question. I will show that, just as in quantum Hall systems, electron interactions allow for a whole new class of states - which we call "fractional topological insulators." These states have excitations with fractional charge and statistics, in addition to protected edge modes.