Semiconductors based on heavy p‑block elements (such as Bi, Sb, Te and Se) that crystallize in the rhombohedral tetradymite structure have long been of interest as the world’s best thermoelectrics (TEs) for room‑temperature operation and have been continuously studied and developed for that application 1 . For simplicity, we clas‑ sify them as ‘tetradymites’ after the mineral of nominal composition, Bi 2 Te 2 S. In the past decade, tetradymites have also emerged as important hosts for the topological surface state, a previously unexplored electronic state of matter. Generally, semiconductors have surface states because the surface represents an interruption of the periodic potentials of the atoms in the bulk (these are Shockley and Tamm states 2 ), which creates additional energy levels that are available to electrons at the surface. In materials with inverted electronic band energies (BOX 1), electrons in the surface states can be topologically protected from back‑scattering and can develop other unusual proper‑ ties. Topological protection provides the surface state electrons with robustness to perturbations that do not break symmetry (time‑reversal symmetry in the case of tetradymites). In materials called topological insulators (TIs), electrical conduction in the bulk is expected to be suppressed, and the topological protection of the surface states strengthened. TIs are semiconductors in which the constituent atoms are heavy and of similar electro‑ negativities; these materials have been reviewed exten‑ sively from the perspective of materials science 3–5 and have also been discussed from a chemical point of view 6,7 . It has often been stated that materials with good TE properties are concomitantly materials with good TI properties 8 . Tetradymites are important materials in both of these areas of materials science (Bi 2 Se 3 has been called ‘the hydrogen atom of TIs’). Although topological physical concepts are novel, exciting and newsworthy, it is topological materials that have turned these concepts into reality and enabled the field to engage experimentalists as well as theoreticians. Two kinds of materials initially displayed topological properties 3–5 , the bismuth–antimony alloy semiconductor Bi 1 − x Sb x ( x ≈ 5 to 20 at%) and thin‑film heterostructures of (Hg,Cd)Te. Both of these materials have limitations that have precluded their widespread experimental study. For bismuth–antimony alloys, the normal surface states that exist on the crystal surfaces in addition to the topological surface states create problems. For (Hg,Cd)Te, the desired effects are displayed only when the material is buried in thin‑film heterostructures, limiting the techniques that can be used to study it. The high vapour pressure of Hg and the sensitivity of the (Hg,Cd)Te properties to the Hg content complicate the control of these properties, such that (Hg,Cd)Te has been fabricated in only a small num‑ ber of laboratories worldwide. By contrast, tetradymites are perfect materials to begin a large, new field of study — the crystals are relatively easy to prepare, the topological physics they display is elegant and new, and the energy dispersions of the surface states are relatively simple 3–5,9,10 . Despite their theoretically ideal behaviour, the tetra‑ dymite bulk materials that were initially studied, Bi 2 Se 3 1 Departments of Mechanical and Aerospace Engineering, Physics, and Materials Science and Engineering, The Ohio State University, Columbus, Ohio 43210, USA. 2 Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA. 3 Department of Physics and Materials Research Institute, the Pennsylvania State University, University Park, Pennsylvania 16802, USA. Correspondence to J.P.H.  heremans.1@osu.edu doi:10.1038/natrevmats.2017.49 Published online 5 Sep 2017 Tetradymites as thermoelectrics and topological insulators Joseph P. Heremans 1 , Robert J. Cava 2 and Nitin Samarth 3 Abstract | Tetradymites are M 2 X 3 compounds — in which M is a group V metal, usually Bi or Sb, and X is a group VI anion, Te, Se or S — that crystallize in a rhombohedral structure. Bi 2 Se 3 , Bi 2 Te 3 and Sb 2 Te 3 are archetypical tetradymites. Other mixtures of M and X elements produce common variants, such as Bi 2 Te 2 Se. Because tetradymites are based on heavy p‑block elements, strong spin‑orbit coupling greatly influences their electronic properties, both on the surface and in the bulk. Their surface electronic states are a cornerstone of frontier work on topological insulators. The bulk energy bands are characterized by small energy gaps, high group velocities, small effective masses and band inversion near the centre of the Brillouin zone. These properties are favourable for high‑efficiency thermoelectric materials but make it difficult to obtain an electrically insulating bulk, which is a requirement of topological insulators. This Review outlines recent progress made in bulk and thin‑film tetradymite materials for the optimization of their properties both as thermoelectrics and as topological insulators. REVIEWS NATURE REVIEWS | MATERIALS VOLUME 2 | ARTICLE NUMBER 17049 | 1 ©2017MacmillanPublishersLimited,partofSpringerNature.Allrightsreserved.