Today there is great diversity in both thinking and practice on this question.
Traditionally, Muslim men may marry women who are of the “People of the Book,” generally defined as Christians and Jews. In this case, a Muslim husband must guarantee the right of his Christian or Jewish wife to worship God according to her religious beliefs.
The reverse, i.e., a Muslim woman marrying a man outside her religion, has traditionally not been allowed, on the grounds that her husband might not guarantee her the right to practice her religion, since he may not to have the same obligation to respect her religion that a Muslim has towards his Christian or Jewish wife. Therefore, for the protection of her freedom of religion, a Muslim woman has traditionally been required to marry a man who will give her the right to practice her faith—that is, a Muslim. This view is based on the patriarchal assumption that the man wields the dominant power in the household and has, therefore, been called into question by some contemporary Muslims, who also cite the absence of a specific text prohibiting such a marriage.