The Bactra Review   The Languages of China
Off the top of my head, and without going into Anatolia or the Caucauses: Albanian, Armenian, Basque, Belorussian, Breton, Bulgarian, Catalan, Czech, Danish, Dutch, English, Estonian, Finnish, Flemish, French, Gaelic, German, (modern) Greek, Italian, Ladino, Lapp, Latvian, Lithuanian, Magyar, Maltese, Norwegian, Polish, Portugese, Provencal, Romanian, Romy (i.e. Gypsy), Romansch, Russian, Ruthenian, Serbo-Croat, Slovak, Slovenian, Spanish (Castillian), Tatar, Turkish, Ukranian, Vlach, Welsh, Yiddish. (German and Italian arguably contain several languages each, e.g. Sardinian in the case of Italian, and a case could be made for Danish and Norwegian being dialects of a single language.) Estonian, Lapp, Finnish and Magyar are Finno-Ugric; Tatar and Turkish are Altaic; Maltese is Semitic; Basque has no known relatives anywhere or at any time; all the rest are Indo-European (Romy coming from India).

Addendum, 1 July 1999: Scott Martens has kindly pointed out a number of languages missing from my list: Corsican, Faroese, Frisian, Gagauz, Gascon, Icelandic, Letzebergish, Pontic Greek, Samoyed, Scots Gaelic, Sorbian and Waloon. (Gagauz is Altaic, Samoyed Finno-Ugric, and the rest, again, Indo-European.) He also informed me that Albanian is really two languages, Gheg and Tosk, that Sinte Romy and Balkans Romy are not mutually intelligible, and that if one variety of Norwegian is part of the same language as Danish, another definitely is not. Another correspondent reminds me that Circassian was still spoken by the descendants of refugees in Ottoman Kosovo in 1900, and that Dalmatian, a Romance language, died out only a little before then. I have not been able to learn whether there were still any speakers of Old Prussian in 1900.