The Tocharian languages, also known as the Arśi-Kuči, Agnean-Kuchean or Kuchean-Agnean languages, are an extinct branch of the Indo-European language family spoken by inhabitants of the Tarim Basin, the Tocharians. The languages are known from manuscripts dating from the 5th to the 8th century AD, which were found in oasis cities on the northern edge of the Tarim Basin and the Lop Desert. The discovery of these languages in the early 20th century contradicted the formerly prevalent idea of an east–west division of the Indo-European language family as centum and satem languages, and prompted reinvigorated study of the Indo-European family. Scholars studying these manuscripts in the early 20th century identified their authors with the Tokharoi, a name used in ancient sources for people of Bactria (Tokharistan). Although this identification is now believed to be mistaken, "Tocharian" remains the usual term for these languages.

The Indo-European languages are a language family native to the overwhelming majority of Europe, the Iranian plateau, and the northern Indian subcontinent. Some European languages of this family—English, French, Portuguese, Russian, Dutch, and Spanish—have expanded through colonialism in the modern period and are now spoken across several continents. The Indo-European family is divided into several branches or sub-families, of which there are eight groups with languages still alive today: Albanian, Armenian, Balto-Slavic, Celtic, Germanic, Hellenic, Indo-Iranian, and Italic; another nine subdivisions are now extinct.