Apokatastasis is a Greek word for restoration, and the doctrine attached to it is the most persistent minority report in Christian history: that hell is real but not final, that its fire corrects rather than merely punishes, and that at the end God is genuinely all in all — every rational creature restored, none permanently lost.
Its great early exponent is Origen of Alexandria, writing around 230. Origen reasoned from the character of God rather than from the vocabulary of judgment: a punishment that accomplishes nothing is unworthy of a physician, and the Greek word rendered "eternal" in the judgment texts, aiōnios, does not straightforwardly mean everlasting but pertains to an age. He combined this with speculations that later proved fatal to his reputation — the pre-existence of souls, a precosmic fall, and a cosmic cycle — and he was careful to present restoration as an inquiry rather than a settled teaching. Gregory of Nyssa, a century and a half later, made the more durable case. He dropped the pre-existence and the precosmic fall entirely and kept the hope, arguing that evil is finite by nature and therefore exhaustible, and that God's purpose in purgation is the removal of evil from the creature rather than the removal of the creature. Isaac of Nineveh, in the Syriac East, put it most directly: it is unfitting to suppose that God did anything, including punish, out of anything but love.
The condemnation is genuinely murky, and the murk is part of the story. A set of anathemas against "Origenism" is associated with the Second Council of Constantinople in 553, but the anathema that addresses restoration appears among nine attributed to the emperor Justinian rather than to the council's own acts, and some scholars place them at a local synod a decade earlier. What is not disputed is that the anathemas target the Origenist package — pre-existent souls, the precosmic fall — rather than universal hope as such, and the strongest evidence for this is Gregory of Nyssa himself: he was never anathematized, in 553 or since, and the same council names him among the holy Fathers it claims to follow.
So the position survived, always at the edge. It runs through Maximus the Confessor's careful ambiguity, through Julian of Norwich's "all shall be well," through Origenist currents in the Reformation radicals, through Schleiermacher, through Karl Barth's refusal to close the question, and into a live contemporary argument in which David Bentley Hart states the case at full strength while most of the tradition still declines it. The atlas holds it as its own framework rather than a footnote to Christian eschatology because it answers the central question differently: the mechanism of judgment is identical, the moral seriousness is identical, and the outcome is that the sentence has a term. Hell, on this reading, is not a destination. It is a procedure, and it ends.