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The evolution and history of the Celtic languages is interesting and somewhat chequered. Celtic languages are descended from the Indo-European stock of languages, which developed into most of the languages that the 'westernized world' speaks today, including English, Spanish, German, French, Russian and so on. The Proto-Celtic language - the first Celtic language that arose from the Indo-European common ancestor - was spoken all over the western continent of Europe. The people who spoke it soon began to die out or to be culturally assimilated by the growing Roman empire (circa 200 BC). The Gauls in fact, were the last known mainland Europeans to speak some form of Celtic. Before all the Celtic speakers were eradicated or integrated, some of them migrated to the British Isles. This signifies the first major division of the Celtic languages.
Now we have Continental Celtic, spoken by the Gauls and other mainlanders, and Insular Celtic, spoken by occupants of the British Isles. The oldest known Insular Celtic language is Old Irish or Goedelic (Gaelic), which eventually became the Irish Gaelic language of today. When settlers moved to what is today mainland Britain, Goedelic became Brythonic. Goedelic and Brythonic are the two major divisions of Celtic that provide the basis for all surviving Celtic languages. When the Irish started to migrate into what is now Scotland, a distinct language, Scottish Gaelic developed. A similar process occurred for those people who inhabited the Isle of Man, who now speak Manx. So there you have the Goedelic Celtic languages: Irish Gaelic, Scottish Gaelic and Manx.
Brythonic, the root for the mainland British languages, also developed into three modern Celtic languages (of which only two survive). Welsh is spoken in Wales and is actually the most widely spoken of the Celtic languages. Cornish also derived from Brythonic Celtic and was spoken in a small region of south-western Britain, and almost died sometime in the 19th Century. However, the language underwent a revival in the latter half of the 20th Century. Breton was the third Brythonic Celtic language to develop.
Breton, although an Insular Celtic language, is actually spoken on continental Europe; in France, to be exact. However, its roots are still from the Brythonic languages. As the history goes, in about 600 AD Britain was invaded by pagan tribes from western Germany, called the Saxons. While the Saxons were sweeping part of southern Britain, some of the natives picked up and left. They landed in north-western France, where their Celtic heritage and language still have an influence on local culture and daily life.