The Qur'an states that marriages between first cousins are allowed. In Sura An-Nisa (4:22–24), Allah mentioned the women who are forbidden for marriage: to quote the Qu'ran, "… Lawful to you are all beyond those mentioned, so that you may seek them with your wealth in honest wedlock…" In Sura Al-Ahzab (33:50),
O Prophet, indeed We have made lawful to you your wives to whom you have given their due compensation and those your right hand possesses from what Allah has returned to you [of captives] and the daughters of your paternal uncles and the daughters of your paternal aunts and the daughters of your maternal uncles and the daughters of your maternal aunts who emigrated with you and a believing woman if she gives herself to the Prophet [and] if the Prophet wishes to marry her, [this is] only for you, excluding the [other] believers. We certainly know what We have made obligatory upon them concerning their wives and those their right hands possess, [but this is for you] in order that there will be upon you no discomfort. And ever is Allah Forgiving and Merciful.
Muslims have practiced marriages between first cousins in all countries since the time of Muhammad. In many countries the most common type is between paternal cousins.
Muhammad actually did marry two relatives. One was a first cousin, Zaynab bint Jahsh, who was not only the daughter of one of his father's sisters but was also divorced from a marriage with Muhammad's adopted son, Zayd ibn Haritha. It was the issue of adoption and not cousinship that caused controversy due to the opposition of pre-Islamic Arab norms. Initially Muhammad had feared to marry Zaynab because of this social disapproval, but later became convinced that the marriage was a duty imposed by revelation from the Quran (Sura Al-Ahzab 33:37).
Many of the immediate successors of Muhammad also took a cousin as one of their wives. Umar (R.A) married his cousin Atikah bint Zayd ibn Amr ibn Nifayl, while Ali (R.A) married Fatimah (R.A), the daughter of his paternal first cousin Muhammad and hence his first cousin once removed.
It has been proposed that the Quranic law of inheritance through the daughter as well as the son encouraged first-cousin marriage in order to keep wealth in the family. Among Islamic societies this rule has not always been followed, but where it was, marrying a first cousin prevented familial wealth from escaping to another clan.