A Muslim man may marry a chaste Christian or Jewish woman. A Muslim woman may marry only a Muslim man. Marriage to people of other faiths is not permitted in the traditional teaching, and some people accept Islam before or after the wedding.
The fuller answer
The reasons given in Islamic law are that in the family, the husband is the guardian and is responsible for the faith of the household, and a Muslim man respects Jesus and Moses, which a non-Muslim husband might not. A person should accept Islam out of conviction, and not to marry, since faith that is pretended is not valid.
Many mixed couples live happily, and the question is dealt with by scholars case by case. Muslims are told to give their children an Islamic upbringing, and any couple should discuss early how they will raise children, and how they will treat each other's family and traditions.
This day [all] good foods have been made lawful, and the food of those who were given the Scripture is lawful for you and your food is lawful for them. And [lawful in marriage are] chaste women from among the believers and chaste women from among those who were given the Scripture before you, when you have given them their due compensation, desiring chastity, not unlawful sexual intercourse or taking [secret] lovers. And whoever denies the faith - his work has become worthless, and he, in the Hereafter, will be among the losers.
And do not marry polytheistic women until they believe. And a believing slave woman is better than a polytheist, even though she might please you. And do not marry polytheistic men [to your women] until they believe. And a believing slave is better than a polytheist, even though he might please you. Those invite [you] to the Fire, but Allah invites to Paradise and to forgiveness, by His permission. And He makes clear His verses to the people that perhaps they may remember.
O you who have believed, when the believing women come to you as emigrants, examine them. Allah is most knowing as to their faith. And if you know them to be believers, then do not return them to the disbelievers; they are not lawful [wives] for them, nor are they lawful [husbands] for them. But give the disbelievers what they have spent. And there is no blame upon you if you marry them when you have given them their due compensation. And hold not to marriage bonds with disbelieving women, but ask for what you have spent and let them ask for what they have spent. That is the judgement of Allah; He judges between you. And Allah is Knowing and Wise.
This answer follows the mainstream Sunni understanding and is general learning, not a fatwa. Where Muslims differ, the answer says so. Have a question of your own? Write to us.