  Topological insulators are electronic materials that have a bulk band gap
like an ordinary insulator, but have protected conducting states on their edge
or surface. The 2D topological insulator is a quantum spin Hall insulator,
which is a close cousin of the integer quantum Hall state. A 3D topological
insulator supports novel spin polarized 2D Dirac fermions on its surface. In
this Colloquium article we will review the theoretical foundation for these
electronic states and describe recent experiments in which their signatures
have been observed. We will describe transport experiments on HgCdTe quantum
wells that demonstrate the existence of the edge states predicted for the
quantum spin Hall insulator. We will then discuss experiments on Bi_{1-x}Sb_x,
Bi_2 Se_3, Bi_2 Te_3 and Sb_2 Te_3 that establish these materials as 3D
topological insulators and directly probe the topology of their surface states.
We will then describe exotic states that can occur at the surface of a 3D
topological insulator due to an induced energy gap. A magnetic gap leads to a
novel quantum Hall state that gives rise to a topological magnetoelectric
effect. A superconducting energy gap leads to a state that supports Majorana
fermions, and may provide a new venue for realizing proposals for topological
quantum computation. We will close by discussing prospects for observing these
exotic states, a well as other potential device applications of topological
insulators.
