Can a Muslim Woman Marry a Non-Muslim Man?

Can a Muslim man marry a Christian or Jewish woman?

It is permissible for a Muslim man to marry a non-Muslim woman if she is Christian or Jewish, but it is not permissible for him to marry a non-Muslim woman who follows any religion other than these two. 

Evidence for the permissibility of marrying a Christian or Jewish woman

The evidence for that is the verse in which Allah says (interpretation of the meaning): 

“Made lawful to you this day are At-Tayyibat [all kinds of Halal (lawful) foods, which Allah has made lawful (meat of slaughtered eatable animals, milk products, fats, vegetables and fruits)]. The food (slaughtered cattle, eatable animals) of the People of the Scripture (Jews and Christians) is lawful to you and yours is lawful to them. (Lawful to you in marriage) are chaste women from the believers and chaste women from those who were given the Scripture (Jews and Christians) before your time when you have given their due Mahr (bridal-money given by the husband to his wife at the time of marriage), desiring chastity (i.e. taking them in legal wedlock) not committing illegal sexual intercourse, nor taking them as girlfriends.” [al-Maidah 5:5] 

Imam al-Tabari said in his commentary on this verse: 

“Chaste women from the believers and chaste women from those who were given the Scripture” means, free woman among those whom have been given the Scripture, namely the Jews and Christians who believe in what is in the Tawrat (Torah) and Injil (Gospel) from among the people who came before you, O believers in Muhammad, whether from among the Arabs or other people; you are permitted to marry them “when you have given their due Mahr (bridal-money given by the husband to his wife at the time of marriage)” which means, if you give to those whom you marry of your (Muslims’) chaste women and their (Jews’ and Christians’) chaste women their mahrs or dowries.” (Tafsir al-Qurtubi, 6/104) 

Can a Muslim man marry a non-Muslim woman?

But it is not permissible for a Muslim man to marry a Magian (Zoroastrian) woman or a communist woman or an idol-worshipping woman, etc. 

The evidence for that is the verse in which Allah says (interpretation of the meaning): 

“And do not marry Al-Mushrikat (idolatresses) till they believe (worship Allah Alone). And indeed a slave woman who believes is better than a (free) Mushrikah (idolatress), even though she pleases you” [al-Baqarah 2:221] 

A mushrikah is an idol-worshipping woman who worships stones, whether from among the Arabs or others. 

Can a Muslim woman marry a non-Muslim man?

It is not permissible for a Muslim woman to marry a non-Muslim man from any other religion, whether from among the Jews or Christians, or any other kafir religion. It is not permissible for her to marry a Jew, a Christian , a Magian, a communist, an idol-worshipper, etc. 

Evidence for the prohibition of marrying a non-Muslim man

The evidence for that is the verse in which Allah says (interpretation of the meaning): 

“And give not (your daughters) in marriage to Al-Mushrikun till they believe (in Allah Alone) and verily, a believing slave is better than a (free) Mushrik (idolater), even though he pleases you. Those (Al-Mushrikun) invite you to the Fire, but Allah invites (you) to Paradise and forgiveness by His Leave, and makes His Ayat (proofs, evidences, verses, lessons, signs, revelations, etc.) clear to mankind that they may remember.” [al-Baqarah 2:221] 

Imam al-Tabari said: 

“What is said concerning the interpretation of the words “And give not (your daughters) in marriage to Al-Mushrikun till they believe (in Allah Alone) and verily, a believing slave is better than a (free) Mushrik (idolater), even though he pleases you” is that what Allah meant by that is that Allah has forbidden the believing women from marrying a mushrik, no matter what kind of shirk he believes in. So, O believers, do not give your daughters in marriage to them, for that is forbidden to you. For you to give them in marriage to a believing slave who believes in Allah and His Messenger and that which he brought from Allah is better for you than to give them in marriage to a free mushrik even if he is of noble descent and honourable origins, even if you like his descent and background…

 It was narrated that Qatadah and al-Zuhri said, concerning the phrase “And give not (your daughters) in marriage to Al-Mushrikun”, It is not permissible for you to give them in marriage to a Jew or a Christian or a mushrik who is not a follower of your religion.” (Tafsir al-Qurtubi, 2/379)

And Allah knows best.