A Muslim is permitted to marry a non-Muslim who falls within the category of ahl al-kitāb (people of the Book); this includes anyone belonging to a religion which received divine scripture. (Quran 5:5) However, it is highly discouraged to do so because cultural and religious differences can give rise to significant marital problems and difficulties, particularly when it comes to raising children. In addition, based on Sharia teachings it is desirable for there to be similarity (kufw) between the couple. (Al-Kāfī, 2: 402)
In respect to marrying other non-Muslims (non ahl al-kitāb), it is not permissible due to non-compatible nature of such unions. (Quran 2:221 and 60:10) However, a jurist can still sanction or validate, with utmost caution, the permissibility of such marriages in instances which are carefully studied and counselled.