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Azeri or Azerbaijani is a Turkic language with about 8 million speakers in Azerbaijan, 32 million speakers in north-western Iran, and a further 6 million speakers in Georgia and Russian Dagestan.
The Arabic script was introduced to the Azerbaijan region in the 7th century and continued to be used to write Azeri until the 1920s. Three different versions of the Arabic script were used during this period: the 28-letter Arabic script, the 32-letter Perso-Arabic script and the 33-letter Turkic Arabic script. None of these was ideal for writing Azeri and various reforms were proposed, particularly during the late 19th and early 20th centuries.
In Iran the Azeri language has always been written with a version of the Arabic script and is know as Azeri Turk.
More info here:
http://www.omniglot.com/writing/azeri.htm