In case of marriage between a Muslim man and a non-Muslim woman must the woman convert before or after the marriage has taken place.

All perfect praise be to Allaah, The Lord of the Worlds.  I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

With regard to the marriage of a Muslim man with a non-Muslim woman, Allaah has forbidden a Muslim to marry idolatress as He Says (what means): {And do not marry idolatresses till they believe (worship Allaah alone). And indeed a slave woman who believes is better than a (free) idolatress, etc, even though she may please you.}[Quran 2:221] The only women who are exempted for non-Muslim men to marry are the chaste women of the People of the Book, that is, the Christians and Jews, as it is permissible for a Muslim to marry them, Allaah Says (what means): {….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 girl-friends. ….}[Quran 5:5]

So it is permissible for a Muslim to marry a chaste woman of the People of the Book.  But if the woman is not chaste, it is not permissible to marry her.

However, it is more appropriate for a Muslim man to marry a Muslim woman for many reasons:

1)     The Muslim woman nurtures her children on the Islamic religion and the children are not harmed by staying with their mother if the husband divorces her or dies.  Conversely, and especially nowadays, the children who stay with a mother from the People of the Book become corrupted because of staying with her after the parents are separated.

2)     It is not certain that the woman of the People of the Book will not influence her children, so they might grow up being affected by the belief and conduct of the polytheists.

3)     It is predominant that the women of the People of the Book at this present time, and especially European women, are not chaste.  Most of them fornicate at an early age.  Being chaste is a condition for the validity of marriage with a woman of the People of the Book. But if one is sure that she is chaste then in principle it is permissible to marry her even if she does not embrace Islam before consummating the marriage.

Allaah Knows best.