Adam as history
Whether Adam is a historical individual through whom death entered (compare Genesis 2-3) or a figurative figure — the exegesis behind creation-and-evolution disputes.
Romans 5:12-19
12Therefore as sin entered into the world through one man, and death through sin; so death passed to all men, because all sinned.
13For until the law, sin was in the world; but sin is not charged when there is no law.
14Nevertheless death reigned from Adam until Moses, even over those whose sins weren’t like Adam’s disobedience, who is a foreshadowing of him who was to come.
15But the free gift isn’t like the trespass. For if by the trespass of the one the many died, much more did the grace of God, and the gift by the grace of the one man, Jesus Christ, abound to the many.
16The gift is not as through one who sinned; for the judgment came by one to condemnation, but the free gift came of many trespasses to justification.
17For if by the trespass of the one, death reigned through the one; so much more will those who receive the abundance of grace and of the gift of righteousness reign in life through the one, Jesus Christ.
18So then as through one trespass, all men were condemned; even so through one act of righteousness, all men were justified to life.
19For as through the one man’s disobedience many were made sinners, even so through the obedience of the one, many will be made righteous.
What each tradition says
Rabbinic / Midrashic
According to The Jewish Encyclopedia, Jewish theologians are divided on the cause of so-called original sin. Some teach it was due to Adam yielding to temptation and eating the forbidden fruit, and has been inherited by his descendants. The majority, however, do not hold Adam responsible for mankind's sins.
Joseph Jacobs (1854-1916) and Judah David Eisenstein (1854-1956), The Jewish Encyclopedia, vol. 11, s.v. 'Sin' (section 'Original Sin') (New York: Funk & Wagnalls, 1901-1906) — Section 'Original Sin' (the article 'Sin', redirected from 'Original Sin')
Patristic
Augustine says death reigned on account of the sin of one man. In that one man all men sinned and were bound by the chain of death, even though they added no sins of their own.
Augustine of Hippo (354-430), On the Merits and Forgiveness of Sins, and on the Baptism of Infants, Book I, in Nicene and Post-Nicene Fathers, ser. 1, vol. 5 (trans. Peter Holmes and Robert Ernest Wallis; Buffalo: Christian Literature Publishing Co., 1887) — Book I, ch. 17 ('Whom Sinners Imitate')
Patristic
John Chrysostom explains “for that all have sinned” as referring to Adam’s fall. When Adam fell, even those who had not eaten of the tree became mortal from him, all of them.
John Chrysostom (c. 349-407), Homilies on the Epistle to the Romans (trans. J. B. Morris and W. H. Simcox; Oxford, 1876), in Nicene and Post-Nicene Fathers, ser. 1, vol. 11 — Homily X, on Romans 5:12
Catholic
Adam's transgression injured not only himself but all his descendants. He lost holiness and justice for us too, and passed on not only death and bodily pains but also sin, the death of the soul. The council declares anathema anyone who denies this, contradicting the apostle: by one man sin entered the world, and death through sin.
Council of Trent (1545-1563); trans. J. Waterworth (1848), The Canons and Decrees of the Sacred and Oecumenical Council of Trent (London: Dolman, 1848) — Session V, Decree on Original Sin, canon 2
Orthodox
John of Damascus states that man was snared by the arch-fiend’s assault and broke his Creator’s command. He was stripped of grace, lost his confidence with God, and was covered with the hardships of a toilsome life.
John of Damascus (c. 675-749), An Exact Exposition of the Orthodox Faith (trans. S. D. F. Salmond), in Nicene and Post-Nicene Fathers, ser. 2, vol. 9 (Edinburgh: T&T Clark, 1898) — Book III, ch. 1 (opening, on the divine economy)
Reformed / Protestant
Calvin holds that all sinned because all are imbued with natural corruption and so became sinful and wicked. He rejects the Pelagian claim that sin descended from Adam to all by imitation, since then Christ would be only the exemplar and not the cause of righteousness.
John Calvin (1509-1564), Commentaries on the Epistle of Paul the Apostle to the Romans (trans. John Owen; Edinburgh: Calvin Translation Society, 1849) — Commentary on Romans 5:12-14
Evangelical
Wesley treats Adam as a figure of the one to come. Adam and the one to come are each public persons and federal heads of mankind. Adam is the fountain of sin and death to mankind by his offense; the other, of righteousness and life by his free gift.
John Wesley (1703-1791), Explanatory Notes upon the New Testament (1755; CCEL's 'Wesley's Notes on the Bible') — Note on Romans 5:14 (within the 5:12-19 argument)