Divorce and remarriage
The exception clause beside Mark and Luke without exception and Paul's abandonment clause in 1 Corinthians 7 — the grounds, permanence and remarriage practice across traditions.
Matthew 19:3-9
3Pharisees came to him, testing him, and saying, “Is it lawful for a man to divorce his wife for any reason?”
4He answered, “Haven’t you read that he who made them from the beginning made them male and female,
5and said, ‘For this cause a man shall leave his father and mother, and shall be joined to his wife; and the two shall become one flesh?’
6So that they are no more two, but one flesh. What therefore God has joined together, don’t let man tear apart.”
7They asked him, “Why then did Moses command us to give her a certificate of divorce, and divorce her?”
8He said to them, “Moses, because of the hardness of your hearts, allowed you to divorce your wives, but from the beginning it has not been so.
9I tell you that whoever divorces his wife, except for sexual immorality, and marries another, commits adultery; and he who marries her when she is divorced commits adultery.”
Mark 10:11-12
11He said to them, “Whoever divorces his wife, and marries another, commits adultery against her.
12If a woman herself divorces her husband, and marries another, she commits adultery.”
What each tradition says
Rabbinic / Midrashic
In the Mishnaic period, the school of Shammai challenged divorce at will, interpreting Deuteronomy 24:1 to mean a husband could not divorce except for sexual immorality. The school of Hillel held that no reason was needed; any act displeasing him entitled him to give a bill of divorce. Hillel's opinion prevailed.
S. S. D. and W. A. (contributors, The Jewish Encyclopedia), The Jewish Encyclopedia, vol. 4, s.v. 'Divorce' (New York: Funk & Wagnalls, 1901-1906) — section 'In the Mishnaic Period'
Patristic
Hermas states that a husband should put his wife away and remain by himself. If he puts his wife away and marries another, he also commits adultery.
Hermas (The Shepherd of Hermas, c. 140-150), The Pastor of Hermas, Mandate IV, ch. 1 (trans. F. Crombie), in Ante-Nicene Fathers, vol. 2 (Buffalo: Christian Literature Publishing Co., 1885) — Mandate IV ('On Putting One's Wife Away for Adultery'), ch. 1
Catholic
The Council of Trent anathematizes anyone who says the Church erred in teaching, in accordance with evangelical and apostolic doctrine, that the bond of matrimony cannot be dissolved for adultery. It adds that neither party, even the innocent, may contract another marriage while the other lives, and that remarriage after putting away an adulterous spouse is adultery.
The Council of Trent (Session XXIV, 11 November 1563), Canons on the Sacrament of Matrimony, Session XXIV (ed. and trans. J. Waterworth, London: Dolman, 1848) — Session XXIV, Canon VII (on the bond's indissolubility)
Orthodox
Basil says the Lord forbids divorce for man and woman alike except for fornication, but custom requires wives to retain husbands even if guilty of fornication. A deserted husband may take another, though Basil holds the woman who takes him guilty of adultery; a wife is not allowed this liberty. A man who deserts an innocent wife may not marry.
Reformed / Protestant
Calvin states that if an unbeliever puts away his wife on account of religion, a brother or sister is freed from the bond of marriage by such rejection. This freedom comes through the unbeliever’s rejection.
John Calvin (1509-1564), Commentary on the Epistles of Paul the Apostle to the Corinthians, vol. 1 (trans. John Pringle, Calvin Translation Society, 1848) — Commentary on 1 Corinthians 7:12-15 (Paul's abandonment clause)
Reformed / Protestant
Calvin states that an exception is added: by fornication the woman cuts herself off as a rotten member from her husband and sets him at liberty. He says those who seek other reasons should be rejected, since they choose to be wise above the heavenly teacher.
John Calvin (1509-1564), Commentary on a Harmony of the Evangelists, Matthew, Mark, and Luke, vol. 2 (trans. William Pringle, Calvin Translation Society, 1846) — Commentary on Matthew 19:3-9 and Mark 10:2-12 (Harmony)
Evangelical
J. C. Ryle says that, with respect to marriage, our Lord teaches the union of husband and wife ought never to be broken off. The exception is the greatest of all causes, namely, actual unfaithfulness.
J. C. Ryle (1816-1900), Expository Thoughts on the Gospels: St. Matthew (1856; CCEL edition) — Exposition of Matthew 19:1-15