Since there are a fair amount of them, languages are grouped below by language family:

**[Basque][1]** - A linguistic isolate native to the Pyrenees mountains between Spain, and France.
[![Basque Map][2]][2]

**[Uralic Languages][3]:**

[![Map of Uralic][4]][12] (click to enlarge)

 - [Finnic Languages][5]: [Finnish][6], Karellian, [Estonian][7], Veps, Ingrian, Votic,
Ludic, Livonian
 - [Hungarian][8]
 - A lot of little Uralic languages near the Urals. (could use fleshing out, but these are all small, and on the fringes of Asia)
 
 https://en.wikipedia.org/wiki/Komi_language

 https://en.wikipedia.org/wiki/Udmurt_language

 https://en.wikipedia.org/wiki/Mari_language

 https://en.wikipedia.org/wiki/Erzya_language

 https://en.wikipedia.org/wiki/Mordvinic_languages

**[Turkic Languages][9]:**

[![Turkic Map][10]][10]

- [Turkish][11] in the portions of the nation of Turkey west of the Bosporus (including Istanbul).

- [Azeri] [13] in the portion of Azerbaijan that is in Europe.
- [Tatar] [19] in Tatarstan area of Russia
- [Kipshak] [20] in Russia and other parts of Eastern Europe
- Kazakh in the Russian-Kazakh border regions

**[Northeast Caucasian (Caspian) Languages][15]:**

[![NE Caucasian Map][18]] (click to open)

- 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][16]:**

[![NW Caucasian Map][17]] (click to open)

- 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][14]:**

- [Georgian][14] in portions of Georgia that someone might consider to be in Europe (or any Georgian speakers over the border in Russia, who are certainly in Europe).

Note that Kartvelian is the most debatable item on this list as the Republic of Georgia is entirely within Asia as its northern border is the ridge of the Caucasus.  On the other hand, many Northeast Caucasian and Northwest Caucasian languages are native tongues north of the ridge of the Caucasus (thus in Europe), so those families indisputably belong on the list.  Also, the northeast corner of Azerbaijan is in Europe, so Azeri belongs on the list (within the Turkic family).

  [1]: https://en.wikipedia.org/wiki/Basque_language
  [2]: https://i.sstatic.net/yG0SI.png
  [3]: https://en.wikipedia.org/wiki/Uralic_languages
  [4]: https://i.sstatic.net/4KWEwm.png
  [5]: https://en.wikipedia.org/wiki/Finnic_languages
  [6]: https://en.wikipedia.org/wiki/Finnish_language
  [7]: https://en.wikipedia.org/wiki/Estonian_language
  [8]: https://en.wikipedia.org/wiki/Hungarian_language
  [9]: https://en.wikipedia.org/wiki/Turkic_languages
  [10]: https://i.sstatic.net/euQFz.png
  [11]: https://en.wikipedia.org/wiki/Turkish_language
  [12]: https://i.sstatic.net/4KWEw.png
  [13]: https://en.wikipedia.org/wiki/Languages_of_Azerbaijan
  [14]: https://en.wikipedia.org/wiki/Kartvelian_languages
  [15]: https://en.wikipedia.org/wiki/Northeast_Caucasian_languages
  [16]: https://en.wikipedia.org/wiki/Northwest_Caucasian_languages
  [17]: https://en.wikipedia.org/wiki/Northwest_Caucasian_languages#/media/File:Northwest_Caucasian_languages_map.png
  [18]: https://en.wikipedia.org/wiki/File:Northeast_Caucasian_languages.png
  [19]: https://en.wikipedia.org/wiki/Tatar_language
  [20]: https://en.wikipedia.org/wiki/Kipchak_languages