According to Jewish law, one may marry one’s niece or cousin unless that is prohibited by civil law.
Rabbi Maurice Lamm writes:
A Man May Marry:
1. His step-sister (a step-parent’s daughter from a previous marriage, even though they were raised together as brother and sister from their earliest youth).
2. His stepfather’s wife (divorced or widowed).
3. The daughter-in-law of his brother or his sister (divorced or widowed).
4. His niece. In American and English civil law, a man may not marry a niece who is the daughter of his brother or sister, but may marry a niece who is the daughter of his wife’s brother or sister. The halakhic permission—even encouragement—to marry the daughter of a brother or sister is superseded by the civil law’s prohibition in this case.
5. His cousin.
6. His stepson’s wife (divorced or widowed).
7. His deceased wife’s sister, but not his divorced wife’s sister (unless she is deceased already).
8. A woman with whom he had relations in their unmarried state.
9. A kohen may marry a widow (who was never divorced).
A Woman May Not Marry:
1. Anyone not of the Jewish faith.
2. The son of an adulterous or incestuous union (mamzer).
3. A married man, until the civil and Jewish divorces have been completed.
4. A man with whom she committed adultery.
5. Her divorced husband, after the death or divorce of her second husband.
6. The following relatives (primary and secondary incest):
(a) Her father, grandfather and ascendants; her stepfather; and the husband of her grandmother and of her ascendants.
(b) Her son, grandson, great grandson; her son-in-law, and the husband of her granddaughter and descendants.
(c) Her husband’s father, or grandfather, and the father of her father-in-law and ascendants; and the father of her mother-in-law.
(d) Her husband’s son or grandson and descendants.
(e) Her brother, half-brother, her full or half-sister’s divorced husband in her sister’s lifetime, and her husband’s brother.
(f) Her nephew.
7. A convert may not marry a kohen.
A Woman May Marry:
1. Her step-brother (a step-parent’s son from a previous marriage, even though they were raised together as brother and sister from their earliest youth).
2. Her step-mother’s former husband (divorced or widowed).
3. The son-in-law of her brother or sister.
4. Her cousin.
5. Her sister’s husband (after her sister’s death, not divorce, unless she is deceased already).
6. Her uncle. In Jewish incest law, an aunt-nephew marriage is prohibited, but an uncle-niece marriage is permitted even though the state prohibits it. A man may marry his deceased wife’s sister, but a woman may not marry her deceased husband’s brother. Even a childless widow, whom the Bible commanded to marry her husband’s brother, must today receive chalitzah, enforced separation.
7. A man with whom she had relations in her unmarried state.
8. A kohen’s daughter does not have the restrictions of a male kohen.
Wikipedia:
Jewish views on incest deal with the sexual relationships which are prohibited by Judaism and rabbinic authorities on account of a close family relationship that exists between persons. Such prohibited relationships are commonly referred to as incest or incestuous, though that term does not appear in the biblical and rabbinic sources. The term mostly used by rabbinic sources is “forbidden relationships in Judaism.”
One of the most notable features of all the lists is that sexual activity between a man and his own daughter is not explicitly forbidden, although the first relation mentioned after the Levitical prohibition of sex with “near kin” is that of “thy father.”[1][2] (This assumes that the Torah is only speaking to men. If it is speaking to everyone, then a woman is not allowed to have sex with her father.[1] It also explicitly prohibits having sex with a woman and her daughter.[1] A man’s daughter is obviously also the daughter of a woman with whom he had sexual relations.) The Talmud argues that this absence is because the prohibition was obvious, especially given the proscription against a relationship with a granddaughter.[3] As with the case of a man’s own daughter, the shortness of the list in Leviticus 20, and especially of that in Deuteronomy, are explained by classical Jewish scholarship as being due to the obviousness of the missing prohibitions.[4][5]
Apart from the case of a man marrying his daughter, the list in Leviticus 18 roughly produces the same rules as were followed in early (pre-Islamic) Arabic culture.[4] However, most tribal nations also disliked exogamous marriage – marriage to completely unrelated people.[4]
Judaism’s view is that prior to the giving of the Torah at Mount Sinai, some of the prohibitions only applied voluntarily. Thus in several prominent cases in the Torah, the incest rules are ignored in favour of marriage to a close relative; Jacob is described as having married his first wife’s sister.
Some secular Biblical scholars have instead proposed that forbidding incest with a daughter was originally in the list, but was then accidentally left out from the copy on which modern versions of the text ultimately depend, due to a mistake by the scribe.
