Topological insulators are electronic materials with a bulk band gap like an ordinary insulator, but with protected conducting states on their edge or surface, protected by spin-orbit interactions and time-reversal symmetry. Two-dimensional topological insulators are quantum spin Hall insulators, closely related to the integer quantum Hall state. Three-dimensional topological insulators support spin-polarized 2D Dirac fermions on their surfaces. This review discusses the theoretical foundation and experimental observations of topological insulators and superconductors. It describes transport experiments on HgTe/CdTe quantum wells that demonstrate edge states of the quantum spin Hall insulator, and experiments on Bi₁₋ₓSbₓ, Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃ that confirm these materials as 3D topological insulators. It also discusses exotic surface states due to induced energy gaps, such as the quantum Hall effect and topological magnetoelectric effect, and Majorana fermions in superconducting proximity. The review concludes with prospects for observing these states and potential applications of topological insulators.Topological insulators are electronic materials with a bulk band gap like an ordinary insulator, but with protected conducting states on their edge or surface, protected by spin-orbit interactions and time-reversal symmetry. Two-dimensional topological insulators are quantum spin Hall insulators, closely related to the integer quantum Hall state. Three-dimensional topological insulators support spin-polarized 2D Dirac fermions on their surfaces. This review discusses the theoretical foundation and experimental observations of topological insulators and superconductors. It describes transport experiments on HgTe/CdTe quantum wells that demonstrate edge states of the quantum spin Hall insulator, and experiments on Bi₁₋ₓSbₓ, Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃ that confirm these materials as 3D topological insulators. It also discusses exotic surface states due to induced energy gaps, such as the quantum Hall effect and topological magnetoelectric effect, and Majorana fermions in superconducting proximity. The review concludes with prospects for observing these states and potential applications of topological insulators.