The Bible teaches that Christ will reign for a period of time before handing the kingdom to his Father:
1 Corinthians 15:24-26
Then comes the end, when he [Christ] delivers the kingdom to God the Father after destroying every rule and every authority and power. 25 For he must reign until he has put all his enemies under his feet. 26 The last enemy to be destroyed is death.
At the end of Revelation, we learn that this period when Christ reigns is to last 1000 years (Revelation 20:1-6).
Then there is a second judgement (Revelation 20:11-13), and death itself is destroyed (Revelation 20:14). After that, only immortal saints will continue to live.
Some people interpret Revelation 21:1 – 22:5 as a description of this period.
And I heard a loud voice from the throne saying, “Behold, the dwelling place of God is with man. He will dwell with them, and they will be his people, and God himself will be with them as their God. 4 He will wipe away every tear from their eyes, and death shall be no more, neither shall there be mourning, nor crying, nor pain anymore, for the former things have passed away.”
Then the angel showed me the river of the water of life, bright as crystal, flowing from the throne of God and of the Lamb through the middle of the street of the city; also, on either side of the river, the tree of life with its twelve kinds of fruit, yielding its fruit each month. The leaves of the tree were for the healing of the nations. No longer will there be anything accursed, but the throne of God and of the Lamb will be in it, and his servants will worship him. They will see his face, and his name will be on their foreheads. And night will be no more. They will need no light of lamp or sun, for the Lord God will be their light, and they will reign forever and ever.
Apart from those chapters, it seems God has chosen not to reveal much to us about what happens at the end of the 1000 years. Just as we don’t know anything about what happened before God created the world, we don’t know much about what happens in this future period either.