The presence in the Supper
'Eat the flesh of the Son of Man' with 'this is my body' — transubstantiation, consubstantiation, spiritual presence, memorial; the presence question across the traditions (also 1 Corinthians 10:16).
John 6:48-58
48I am the bread of life.
49Your fathers ate the manna in the wilderness and they died.
50This is the bread which comes down out of heaven, that anyone may eat of it and not die.
51I am the living bread which came down out of heaven. If anyone eats of this bread, he will live forever. Yes, the bread which I will give for the life of the world is my flesh.”
52The Jews therefore contended with one another, saying, “How can this man give us his flesh to eat?”
53Jesus therefore said to them, “Most certainly I tell you, unless you eat the flesh of the Son of Man and drink his blood, you don’t have life in yourselves.
54He who eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day.
55For my flesh is food indeed, and my blood is drink indeed.
56He who eats my flesh and drinks my blood lives in me, and I in him.
57As the living Father sent me, and I live because of the Father; so he who feeds on me, he will also live because of me.
58This is the bread which came down out of heaven—not as our fathers ate the manna, and died. He who eats this bread will live forever.”
1 Corinthians 11:23-29
23For I received from the Lord that which also I delivered to you, that the Lord Jesus on the night in which he was betrayed took bread.
24When he had given thanks, he broke it and said, “Take, eat. This is my body, which is broken for you. Do this in memory of me.”
25In the same way he also took the cup, after supper, saying, “This cup is the new covenant in my blood. Do this, as often as you drink, in memory of me.”
26For as often as you eat this bread and drink this cup, you proclaim the Lord’s death until he comes.
27Therefore whoever eats this bread or drinks the Lord’s cup in a way unworthy of the Lord will be guilty of the body and the blood of the Lord.
28But let a man examine himself, and so let him eat of the bread, and drink of the cup.
29For he who eats and drinks in an unworthy way eats and drinks judgment to himself if he doesn’t discern the Lord’s body.
What each tradition says
Rabbinic / Midrashic
Kohler holds it probable that the institution developed out of Essene communion-meals and was only later referred to Jesus. The Essene communion-feasts, borrowed from Parseeism, retained the hope for paradise banquets of leviathan; this is why Jesus is reported as especially referring to wine in the Kingdom of God.
Kaufmann Kohler (1843-1926), The Jewish Encyclopedia, vol. 8, s.v. 'Lord's Supper' (New York: Funk & Wagnalls, 1901-1906) — article body (signed K.)
Patristic
Ignatius of Antioch states that the Eucharist is the flesh of our Savior Jesus Christ. That flesh suffered for our sins, and the Father, of His goodness, raised it up again.
Ignatius of Antioch (c. 35-c. 107), Epistle to the Smyrnaeans, ch. 7, in Ante-Nicene Fathers, vol. 1 (ed. Alexander Roberts and James Donaldson; Buffalo: Christian Literature Publishing Co., 1885) — ch. 7, 'Let us stand aloof from such heretics'
Catholic
The Council of Trent declares that in the sacrament of the most holy Eucharist are contained truly, really, and substantially the body and blood, together with the soul and divinity, of our Lord Jesus Christ, and therefore the whole Christ. It anathematizes anyone who says he is therein only as a sign, figure, or virtue.
The Council of Trent, Session XIII (1551), The Canons and Decrees of the Sacred and Oecumenical Council of Trent (trans. J. Waterworth; London: Dolman, 1848) — Session XIII, canons on the Most Holy Sacrament of the Eucharist, canon 1
Orthodox
The Synod of Jerusalem believes the Lord Jesus Christ is present truly and really, not typically, figuratively, by superabundant grace, by a bare presence, or by impanation as Luther’s followers suppose. After the consecration of the bread and wine, the bread is transmuted.
The Synod of Jerusalem (1672), under Patriarch Dositheus, The Acts and Decrees of the Synod of Jerusalem, sometimes called the Council of Bethlehem, holden under Dositheus, Patriarch of Jerusalem, in 1672 (trans. J. N. W. B. Robertson; London: Thomas Baker, 1899) — Confession of Dositheus, Decree XVII
Reformed / Protestant
The Westminster Confession says believers by faith really receive and feed upon Christ crucified and all benefits of his death, yet not carnally or corporally but spiritually. Christ's body and blood are not carnally in, with, or under the bread and wine, but are as really, though spiritually, present to believers' faith as the elements are to their outward senses.
The Westminster Assembly (1647), The Westminster Confession of Faith, in Philip Schaff (ed.), The Creeds of Christendom, vol. 3 (New York: Harper & Brothers, 1877; CCEL text) — ch. XXIX ('Of the Lord's Supper'), §7
Evangelical
Albert Barnes states that 1 Corinthians 11:26 is a direct and positive refutation of the doctrine of the papists, which says the bread is changed into the real body of the Lord Jesus. He notes that here it is expressly called bread—bread still—after the consecration.
Albert Barnes (1798-1870), Notes on the New Testament, Explanatory and Practical (ed. Robert Frew; Grand Rapids: Baker Book House, 1949-50 reprint of the 19th-c. text, CCEL) — Note on 1 Corinthians 11:26