“In doing this he acted very well and honorably, taking account of the resurrection. For if he were not expecting that those who had fallen would rise again, it would have been superfluous and foolish to pray for the dead. … Therefore he made atonement for the dead, that they might be delivered from their sin.”
Short answer
The Church teaches that all who die in God's grace are assured of heaven. Those who need no purification are with Christ immediately after death. Those who die in grace but still imperfectly purified are cleansed first, and this cleansing is entirely different from the punishment of the damned. Christians of the early centuries prayed and offered the Eucharist for their dead, a practice that supposes the dead can be helped.
Drafted from primary sources. Every quotation is checked against the page it links to.
Where this teaching appears on the Offshoots timelines: January 1523, No purgatory; 1895, Purgatorial fire rejected, prayer for the dead kept.
16 quotations in date order
“And whoever says a word against the Son of man will be forgiven; but whoever speaks against the Holy Spirit will not be forgiven, either in this age or in the age to come.”
“each man's work will become manifest; for the Day will disclose it, because it will be revealed with fire, and the fire will test what sort of work each one has done. If the work which any man has built on the foundation survives, he will receive a reward. If any man's work is burned up, he will suffer loss, though he himself will be saved, but only as through fire.”
“But nothing unclean shall enter it, nor any one who practices abomination or falsehood, but only those who are written in the Lamb's book of life.”
“But I trusted that my prayer would bring help to his suffering; and I prayed for him every day until we passed over into the prison of the camp, … I saw that that place which I had formerly observed to be in gloom was now bright; and Dinocrates, with a clean body well clad, was finding refreshment. … Then I understood that he was translated from the place of punishment.”
“Indeed, she prays for his soul, and requests refreshment for him meanwhile, and fellowship (with him) in the first resurrection; and she offers (her sacrifice) on the anniversaries of his falling asleep.”
“The bishops our predecessors religiously considering this, and wholesomely providing for it, decided that no brother departing should name a cleric for executor or guardian; and if any one should do this, no offering should be made for him, nor any sacrifice be celebrated for his repose. For he does not deserve to be named at the altar of God in the prayer of the priests, who has wished to call away the priests and ministers from the altar.”
“Then on behalf also of the Holy Fathers and Bishops who have fallen asleep before us, and in a word of all who in past years have fallen asleep among us, believing that it will be a very great benefit to the souls, for whom the supplication is put up, while that holy and most awful sacrifice is set forth.”
“Let us then give them aid and perform commemoration for them. For if the children of Job were purged by the sacrifice of their father, why do you doubt that when we too offer for the departed, some consolation arises to them?”
“Nor can it be denied that the souls of the dead are benefited by the piety of their living friends, who offer the sacrifice of the Mediator, or give alms in the church on their behalf. But these services are of advantage only to those who during their lives have earned such merit, that services of this kind can help them. For there is a manner of life which is neither so good as not to require these services after death, nor so bad that such services are of no avail after death;”
“But temporary punishments are suffered by some in this life only, by others after death, by others both now and then; but all of them before that last and strictest judgment. But of those who suffer temporary punishments after death, all are not doomed to those everlasting pains which are to follow that judgment;”
“According to the general disposition of God, the souls of all the saints who departed from this world before the passion of our Lord Jesus Christ and also of the holy apostles, martyrs, confessors, virgins and other faithful who died after receiving the holy baptism of Christ- provided they were not in need of any purification when they died, or will not be in need of any when they die in the future, or else, if they then needed or will need some purification, after they have been purified after death … all these souls, immediately (mox) after death and, in the case of those in need of purification, after the purification mentioned above, since the ascension of our Lord and Saviour Jesus Christ into heaven, already before they take up their bodies again and before the general judgment, have been, are and will be with Christ in heaven, in the heavenly kingdom and paradise, joined to the company of the holy angels.”
“Also, if truly penitent people die in the love of God before they have made satisfaction for acts and omissions by worthy fruits of repentance, their souls are cleansed after death by cleansing pains; and the suffrages of the living faithful avail them in giving relief from such pains, that is, sacrifices of masses, prayers, almsgiving and other acts of devotion which have been customarily performed by some of the faithful for others of the faithful in accordance with the church's ordinances.”
“Whereas the Catholic Church, instructed by the Holy Ghost, has, from the sacred writings and the ancient tradition of the Fathers, taught, in sacred councils, and very recently in this oecumenical Synod, that there is a Purgatory, and that the souls there detained are helped by the suffrages of the faithful, … but principally by the acceptable sacrifice of the altar; … While those things which tend to a certain kind of curiosity or superstition, or which savour of filthy lucre, let them prohibit as scandals and stumbling-blocks of the faithful.”
“All who die in God's grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven. … The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned.”
“His gaze, the touch of his heart heals us through an undeniably painful transformation “as through fire”. But it is a blessed pain, in which the holy power of his love sears through us like a flame, enabling us to become totally ourselves and thus totally of God. … our defilement does not stain us for ever if we have at least continued to reach out towards Christ, towards truth and towards love. Indeed, it has already been burned away through Christ's Passion.”