Overview
Since there are a fair amount of them, languages are grouped below by language family:
Basque
A linguistic isolate native to the Pyrenees mountains between Spain, and France.
Source: "Location of the Basque-language provinces within Spain and France" by Eddo from Wikipedia.org
Uralic Languages
Source: "Linguistic maps of the Uralic languages" by Eddo derived based on a work by Chumwa from Wikipedia.org
Finnic Languages: Finnish, Karelian, Estonian, Vepsian, Ingrian, Votic, Ludic, Livonian
A lot of little Uralic languages near the Urals. (likely incomplete, but these are all small, and on the fringes of Asia)
Afro-Asiatic Languages:
- Maltese*, spoken on the island of Malta. Quite closely related to Arabic, this is the only Afro-Asiatic language that is an official language of an EU member country.
Turkic Languages
Source: "An accurate representation of the areas in which Turkic languages are spoken." Copyright by Mirza Farahani, used under CC BY-SA 4.0 from wikipedia.org.
Turkish in the portions of the nation of Turkey west of the Bosporus (including Istanbul).
Azeri in the portion of Azerbaijan that is in Europe.
Tatar in Tatarstan area of Russia
Kipshak in Russia and other parts of Eastern Europe
Bashkir language is a Turkic language belonging to the Kipchak branch. It is co-official with Russian in the Republic of Bashkortostan, European Russia
Kazakh in the Russian-Kazakh border regions
Gagauz language by the Gagauz people of Moldova, Ukraine, Russia, and Turkey, and it is the official language of the Autonomous Region of Gagauzia in Moldova.
Chuvash language in European Russia, primarily in the Chuvash Republic and adjacent areas.
Caucasian language families
These three language families are not considered to be related to each other, so this a geographic grouping, not a linguistic one. The below language families are all native to the region between the Black and Caspian seas.
Northeast Caucasian (Caspian) Languages
Source: "Approximate distribution of the branches of the Northeast Caucasian languages" by JorisvS from wikipedia.org
Spoken in both Azerbaijan and in the Russian Republics of Dagestan, Chechnya and Ingushetia. These include Chechen, Avar, Lezgian, Dargwa, Ingush, Lak, and Nakh.
Northwest Caucasian (Pontic) Languages
Source: "Approximate distribution of the branches of the Northwest Caucasian languages" by Gaga.vaa from wikipedia.org
Within Europe, spoken primarily in the Russian Republics of Adygea, Kabardino-Balkaria, Karachay–Cherkessia. Members of the family represented in Europe include Karbardian and Adyghe.
Kartvelian (Iberian) Languages
- Georgian in European portions of Georgia*
Mongolian languages
In the form of Kalmyckian Oirat, with Kalmyckia also the region in Europe with Buddhism as the main religion.
Footnotes:
* - Geographically debatable