The disputed theory centers on the presumed mass conversion to Judaism of the Khazars — an extinct multiethnic kingdom that included Iranians, Turks, Slavs and Circassians — in the eighth century. Largely unsupported by genetic studies, it is popular in anti-Semitic and anti-Zionist literature because it is seen to dispel the notion of Jewish peoplehood, presenting it as a hoax.
There is great genetic similarity between Ashkenazim and the Jews of Rome, who came from the Land of Israel and later from the Mediterranean. In no way can the same explanation of the origins of the Jews in Europe (the Khazar theory) apply to the Jews of Rome.