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One of the biggest language families in Europe is the romance language family. French, Spanish, Portuguese, Romanian and Italian all belong to the romance family. These languages are descended from Latin.
Another important family is Germanic, with members like English, Dutch, German and Swedish. Gothic was also a member of this family, but this language died out.
Further East we have the Slavic language family with Russian, Polish and Czech for instance.
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A smaller family is Celtic, the family which Irish, Scots, Gaelic, Welsh and Breton belong to.
These families are in fact all members of a bigger, older language family called Indo-European. These go back further to a common law spoken in central Asia.
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Another family is Finno-Ugric with languages is like Finnish and Hungarian. The language of the Sami people of Finland also belongs here.
Basque is the odd one out in that it doesn't seem to have any relatives at all! It is spoken in Northern Spain and Southern France in what we call the Basque country.