Similar to the New Testament writers (a post for another day), the Early Church understood that they were living in the last days. For example, Ignatius wrote “these last times” (Gr. έσχατοι καιροί, Ign. Eph. 11.1) and that Jesus Christ “at the end appeared” (Gr. ἐν τέλει ἐφάνη, Ign. Magn. 6.1). Clement and Barnabas describe their present moment as “the last days” (Gr. ἐσχάτων τῶν ἡμέρῶν, 2 Clem 14:2; Barn. 12:9; 16:5) and that Christians through Jesus have been given the firstfruits of the taste of things future (Gr. τῶν μελλόντων δοuς ἀπαρχaς ἡμῖν γεύσεως, Barn. 1.7). Thus, the Apostolic Fathers followed the NT writers in the inaugurated aspect of the end times. In fact, some writers had such a heightened notion of inaugurated eschatology that they compared their present Christian experience to the blessings of the Garden of Eden (Diogn. 12), and possibly with this view, the “seer” in the Shepherd of Hermas even asks the question of whether “the consummation had already arrived” (Herm. Vis. 3.8.9).
Also following the NT writers, the Early Church connected the latter-day new creation theme with their present Christian experience. Ignatius explained that Christ introduced “the newness of eternal life” (Ign. Eph. 19.3) by himself becoming a “new man” (Ign. Eph. 20.1) as a result of his resurrection. According to the Early Church, resurrection occurs first spiritually in the present and is the means by which people enter into the new creation. This inaugurated resurrection of Christians is due to their participation with Christ’s resurrection (Ign. Magn. 9; Ign. Trall. 9.2. Cf. 1 Clem. 24.1; Pol. Phil. 2.2; Barn. 5.6-7).
Furthermore, the Early Church had similar views as the NT writers regarding their present tribulation as a sign of the inaugurated last days. Barnabas refers to “the deception of the present age” (Gr. τnν πλάνην τοῦ νῦν καιροῦ, Barn. 4.1) and understands that the present vantage point is that people are living “in the age of lawlessness” (Gr. ἐν τῷ ἀνόμῳ καιρῷ, Barn, 4.9). The Shepherd of Hermas mentions that “great tribulations” (Gr. μεγάλας θλίψεις, Herm. Vis. 2.7.1) have already been endured which apparently prepares Christians to “endure the coming great tribulation” (Gr. ὑπομένετε τnν θλῖψιν τnν ἐρχομένην τnν μεγάλην, Herm. Vis. 2.7.1).
While believing in the present inaugurated latter days, early christian writers believed in the “age to come” (Gr. o αἰων o ἐπερχόμεός, Herm. Vis. 4.3.5). Early Christian writers and teachers such as Justin Martyr, Irenaeus, and Tertullian followed the NT writers in their belief of the future consummated resurrection of the body when Christ’s returns in his resurrected body. Similar to the NT writers, early Christian writers believed that other events (i.e. great tribulation, appearance of the antichrist, judgement) would be linked with Christ’s return. Hippolytus wrote, “When he [antichrist] makes war against the saints, and persecutes them, then we may expect the manifestation of the Lord from heaven.” Justin Martyr wrote, “He [Jesus] shall come from heaven with glory, when the man of apostasy, who speaks strange things against the most High, shall venture to do unlawful deeds on the earth against Christians.” Thus, the Early Church mostly agreed that the return of Christ would come after the period of great tribulation and would involve the resurrection of believers.
The Early Church emphasized the literal interpretation of Revelation 20:1-6 which describes the reign of Christ upon the earth for one thousand years (millennium). Papias wrote, “there will be a period of a thousand years after the resurrection from the dead when the kingdom of Christ will be set up in material form on this earth.” Justin Martyr wrote, “I and others who are right-minded Christians on all points, are assured that there will be a resurrection of the dead and a thousand years in Jerusalem, which will then be built, adorned and enlarged.” Irenaeus and Tertullian held similar views concerning the millennium; thus, suffice it to say, many influential Early Church writers held to what has been called the “historic premillennialism” view—Christ returns, Christians bodily resurrected, Christians reign with Christ on earth for one thousand years.
Like the NT writers, the Early Church believed in a future final judgement after Christ’s return. The Athanasian Creed summarized this belief by explaining that Christ “will come to judge the living and the dead. At his coming all people will rise again with their bodies and will give account for their own works. Those who have done good will go into eternal life, and those who have done evil will go into eternal fire.” Athenagoras and Hippolytus further explained that the “eternal fire” for those who have done evil does not equate to annihilationism—that people would cease to exist, but rather the final state for them is that of a conscious eternal punishment.
Lastly, the Early Church followed the NT writers hope of the new heavens and the new earth in the future. However, there was some disagreement within the Early Church about how this would take place. Irenaeus believed that a divine renewal of the existing heavens and earth would occur, whereas, Tertullian and Melito of Sardis believed in a total annihilation of the current heavens and earth before the creation of the new. Therefore, along with their “historic premillennialism” views, the Early Church believed that unbelievers will be resurrected after the one thousand year reign, followed by final judgment and then God establishes the eternal state of heaven and hell.
 G.K. Beale, “Eschatology,” in Dictionary of the Later New Testament and Its Developments (DLNT), edited by R. P. Martin and P. H. Davids (Downers Grove, IL: InterVarsity Press, 1997), 341.  Beale, “Eschatology,” in DLNT, 341.  Beale, “Eschatology,” in DLNT, 341.  Beale, “Eschatology,” in DLNT, 342.  Beale, “Eschatology,” in DLNT, 342  T.A. Noble, “Eschatology,” in New Dictionary of Theology (NDT): Historical and Systematic, edited by Martin Davie et al. 2nd ed. (Downers Grove, Illinois: IVP Academic, 2016), EPUB edition.  Hippolytus, Fragments from Commentaries: Daniel, 2.7, in ANF, 5:179 quoted by Gregg R. Allison, Historical Theology : An Introduction to Christian Doctrine : A Companion to Wayne Grudem’s Systematic Theology (Grand Rapids, MI: Zondervan, 2011), EPUB edition, pt.7, ch. 31, “Eschatological Beliefs in the Early Church.”  Justin Martyr, Dialogue with Trypho, 110, in ANF, 1:253-54 quoted by Allison, Historical Theology, EPUB edition, pt.7, ch. 31, “Eschatological Beliefs in the Early Church.”  Allison, Historical Theology, EPUB edition, pt.7, ch. 31, “Eschatological Beliefs in the Early Church.”  Fragments of Papias 3.12 quoted by Allison, Historical Theology, EPUB edition, pt.7, ch. 31, “Eschatological Beliefs in the Early Church.”  Justin Martyr, Diaologue with Trypho, 80, in ANF, 1:239 quoted by Allison, Historical Theology, EPUB edition, pt.7, ch. 31, “Eschatological Beliefs in the Early Church.”  Allison, Historical Theology, EPUB edition, pt.7, ch. 31, “Eschatological Beliefs in the Early Church” citing Irenaeus, Against Heresies, 5.32-36, in ANF, 1:561-67.  Athanasian Creed, 40-43, in Schaff, 2:69-70 cited by Allison, Historical Theology, EPUB edition, pt.7, ch. 32 “Judgment and Punishment in the Early Church.” Allison, Historical Theology, EPUB edition, pt.7, ch. 32 “Judgment and Punishment in the Early Church.” Allison, Historical Theology, EPUB edition, pt.7, ch. 33, “The Idea of the Eternal State in the Early Church.”  Allison, Historical Theology, EPUB edition, pt.7, ch. 31, “Eschatological Beliefs in the Early Church” citing Irenaeus, Against Heresies, 5.32-36, in ANF, 1:561-67.