The new covenant
Who the new covenant is made with and whether it is fulfilled in the church, in Israel, or both.
Jeremiah 31:31-34
31“Behold, the days come,” says Yahweh, “that I will make a new covenant with the house of Israel, and with the house of Judah:
32not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which covenant of mine they broke, although I was a husband to them,” says Yahweh.
33“But this is the covenant that I will make with the house of Israel after those days,” says Yahweh: “I will put my law in their inward parts, and I will write it in their heart. I will be their God, and they shall be my people.
34They will no longer each teach his neighbor, and every man teach his brother, saying, ‘Know Yahweh;’ for they will all know me, from their least to their greatest,” says Yahweh: “for I will forgive their iniquity, and I will remember their sin no more.”
Hebrews 8:8-13
8For finding fault with them, he said, “Behold, the days come”, says the Lord, “that I will make a new covenant with the house of Israel and with the house of Judah;
9not according to the covenant that I made with their fathers, in the day that I took them by the hand to lead them out of the land of Egypt; for they didn’t continue in my covenant, and I disregarded them,” says the Lord.
10“For this is the covenant that I will make with the house of Israel. After those days,” says the Lord; “I will put my laws into their mind, I will also write them on their heart. I will be their God, and they will be my people.
11They will not teach every man his fellow citizen, and every man his brother, saying, ‘Know the Lord,’ for all will know me, from their least to their greatest.
12For I will be merciful to their unrighteousness. I will remember their sins and lawless deeds no more.”
13In that he says, “A new covenant”, he has made the first old. But that which is becoming old and grows aged is near to vanishing away.
What each tradition says
Rabbinic / Midrashic
Radak says the covenant’s newness is that it will be kept, not broken like the covenant b’nei Yisrael cut with God at Mount Sinai. He associates this with Yirmiyahu’s prophecy of a “new Torah” that will come in the future.
Radak (R. David Kimhi), Radak on Jeremiah 31:31 — Sefaria Community Translation, served by Sefaria — Commentary on Jeremiah 31:31
Patristic
Chrysostom states that the first covenant came to an end. He says this was shown from the Priest and is now shown more clearly by express words that it has been cast out.
John Chrysostom, NPNF1-14. Saint Chrysostom: Homilies on the Gospel of St. John and the Epistle to the Hebrews — Homily XIV on Hebrews (on Hebrews 8:8-13)
Catholic
Haydock states that the covenant made with the captives was not this new covenant. Their covenant had grown old and come to an end, as St. Paul shows in Hebrews 8:8. They were not divided as they had been, according to Ezekiel 37:16.
George Leo Haydock, George Haydock's Catholic Bible Commentary — Commentary on Jeremiah 31:31
Orthodox
No sourced position found. Checked Philaret's Longer Catechism (pravoslavieto.com) — 'new covenant' appears only in Q. 27-28 (the Old/New Testament division), with no comment on Jer. 31:31 or Heb. 8:8-13; CCEL NPNF (Eastern fathers): the only public-domain English comment on the passage, Chrysostom's Hom. XIV on Hebrews, is already used for the patristic row; archive.org searches for a public-domain English Orthodox commentary on Jeremiah returned nothing on this passage.
Reformed / Protestant
Calvin says this passage necessarily refers to the kingdom of Christ. He explains that without Christ the people could not or ought not have hoped for anything superior to the Law, since the Law was a rule of the most perfect doctrine.
John Calvin, Commentary on Jeremiah and Lamentations - Volume 4 — On Jeremiah 31:31-32
Evangelical
Wesley says the new covenant is made not only with the Jews but with all who are ingrafted into that olive. It is not called new because it was new in substance, for it was made with Abraham and with the Jews, but because it was revealed after a new manner, more fully, particularly, plainly, and clearly.
John Wesley, Explanatory Notes upon the Old Testament — Note on Jeremiah 31:31