Topological insulators are nonmagnetic band insulators in which spin-orbit coupling acts in roughly similar fashion to the magnetic field in the integer quantum Hall effect. These materials are insulating except at boundaries, where there are protected edge or surface states that may have been detected in recent experiments on heavy semiconductors at Wuerzburg and Princeton. While the two-dimensional topological insulator is closely connected to the integer quantum Hall effect, the strong topological insulator in three dimensions is more subtle and has several novel properties, including a protected surface state with Dirac-like excitations as in graphene. This talk reviews the theoretical understanding of these states, discusses recent experimental progress, and (as time permits) presents some new work on a three-dimensional topological insulator state in correlated electron materials.