What is the Catholic understanding of the thousand-year reign of Christ and his people?
What is the Catholic biblical interpretation and what have the Church Fathers said on the matter?
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A caveat about my answer below is that I'm a Lutheran (LCMS) pastor and am trying to do justice to the Catholic view on this. Lutherans, Catholics, Orthodox and others actually share the same view on the "thousand-year reign."
The Catholic view of the thousand-year reign is termed amillennialism. As I mentioned, this view is shared by Lutherans, Orthodox, and others. What it basically means is that the thousand-year reign is symbolic of the period from Christ's ascension to his return. That is to say, Christ and his people reign now in the New Testament Church, since Christ has already defeated sin, death, and the devil through his death and resurrection. When Christ returns, these enemies of creation will be cast out and God will again dwell directly with his people. In the current reign, God dwells among his people through His Sacraments; at Christ's return, God will dwell directly with His people "immediately" (that is, not "mediated" through the Sacraments - the exact definition and number of Sacraments varies among the Catholic, Lutheran, and Orthodox views).
One implication of this view is that the Book of Revelation covers the entire span from Christ's ascension to his return and is not specifically about "the end." For example, Revelation 5 is often seen as "heaven's view" of Christ's ascension, whereas Acts 1 is "earth's view." The point of Revelation is that the Lord is with His Church, even as it is battered and killed on earth, and he is returning to resurrect His people to dwell with Him on the restored, new heaven and earth (see, for example, Pope Francis' remarks at: http://w2.vatican.va/content/francesco/en/audiences/2014/documents/papa-francesco_20141126_udienza-generale.html )
A good Wikipedia article comparing views of the millennium is at: http://en.wikipedia.org/wiki/Amillennialism
A Catholic discussion of millennialism (as opposed to amillennialism), with some historical notes concerning the Church Fathers, is at: http://www.newadvent.org/cathen/10307a.htm