Topological insulators are electronic materials that exhibit a bulk band gap like ordinary insulators but have protected conducting states on their edges or surfaces. These states arise from the combination of spin-orbit interactions and time-reversal symmetry. The 2D topological insulator, known as a quantum spin Hall insulator, shares similarities with the integer quantum Hall state. A 3D topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. This article reviews the theoretical foundation for topological insulators and superconductors, and describes recent experiments that have observed the signatures of these materials. Transport experiments on HgTe/CdTe quantum wells demonstrate the existence of edge states predicted for the quantum spin Hall insulator. Experiments on materials like Bi1−xSbx, Bi2Se3, Bi2Te3, and Sb2Te3 establish these materials as 3D topological insulators and directly probe the topology of their surface states. The article also discusses exotic states that can occur at the surface of a 3D topological insulator due to induced energy gaps, such as a magnetic gap leading to a novel quantum Hall state and a superconducting energy gap leading to Majorana fermions. These exotic states have potential applications in spintronics and topological quantum computation.Topological insulators are electronic materials that exhibit a bulk band gap like ordinary insulators but have protected conducting states on their edges or surfaces. These states arise from the combination of spin-orbit interactions and time-reversal symmetry. The 2D topological insulator, known as a quantum spin Hall insulator, shares similarities with the integer quantum Hall state. A 3D topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. This article reviews the theoretical foundation for topological insulators and superconductors, and describes recent experiments that have observed the signatures of these materials. Transport experiments on HgTe/CdTe quantum wells demonstrate the existence of edge states predicted for the quantum spin Hall insulator. Experiments on materials like Bi1−xSbx, Bi2Se3, Bi2Te3, and Sb2Te3 establish these materials as 3D topological insulators and directly probe the topology of their surface states. The article also discusses exotic states that can occur at the surface of a 3D topological insulator due to induced energy gaps, such as a magnetic gap leading to a novel quantum Hall state and a superconducting energy gap leading to Majorana fermions. These exotic states have potential applications in spintronics and topological quantum computation.